Patton Oswalt Recommended Psycho Goreman on his Twitter feed. And holly crap, I’m SO glad I watched it. It’s like a demented version of Power Rangers.
Lot’s of “WTF” and completely hilarious. Rated PG but it does have lots of gore. But the gore is so unrealistic, I don’t think it would be too bad for the kiddos depending on their personality type.
Seems like the kids have more sway on the selection of the movies. And it also seems they are at prime age for the Harry Potter movies (all 27 of them 
If they’ve seen all the other Star Wars movies, I’d push for “Rogue One”. I thought this was by far the best of the latest batch (but I did not see “Solo”).
I think “John Carter” (of Mars) would be a good call. This has been panned by most, but I enjoyed it.
I also think some sci-fi that is more realistic might be good: “The Martian”.
Although I love “Galaxy Quest”, and it is one of my favorite movies, I think if you don’t know Star Trek most of the jokes would be lost.
Spaceballs (PG, 1987) – available free via Pluto and YouTube; rent via Amazon Prime.
Robin Hood: Men in Tights (PG-13, 1993) – available via Amazon Prime.
Ladyhawke (PG-13, 1985) – available via Amazon Prime.
OK, I hadn’t realized Explorers was that … goofy. 
Definitely will consider.
Thanks again to all for the suggestions. I’ll stop replying to every one of them individually, but rest assured I am reading all of them and giving them their due consideration.
Time Bandits I still sometimes have nightmares about and I’m 44.
I’m in 100% agreement about the Iron Giant. I will keep suggesting it periodically no matter how many times they shoot it down. Maybe I’ll wear them down eventually. That tactic worked for my wife with Mighty Ducks.
Does Willow hold up? I don’t remember much about it, though I did enjoy it at the time. But I wonder how much that was being starved for a good fantasy movie in a pre-LOTR world.
Believe me, I’ve tried. My screen name is GONZOron for a reason. I don’t understand their objection to all things Muppet. (The little one even likes the new version of Muppet Babies, but has no interest in the real Muppets).
It’s been considered. But the hope has always been that with the vast world of film history, much of which is available at the push of a button, we can find something that everyone will enjoy, or at least tolerate. The more difficult that becomes, the more we consider going toward some kind of rotating choosing method. But the goal is to have a relaxing shared experience as a family, not to force anyone to watch something they don’t want (“This is a classic, you’ll watch it and you’ll like it!”), nor to have everyone go off and do their own thing like we do for the rest of the week.
And though I may have portrayed it that way, it’s not always the kids vetoing. I have put my foot down for Monster High, and my wife and I were the ones who objected to Tinkerbell.
Definitely open to other genres, though I’m not sure about Ernesto’s Manifesto. Seems like some adult themes there with the bikini girls lining up to kiss him??
I’d love to find a good comedy that’s more grown up than say the Wimpy Kid series or Freaky Friday, but less “adult” than something like See No Evil, Hear No Evil, Blazing Saddles, or Clue. (and nowhere near something like Clerks, Harold & Kumar, or Borat) (all movies I love, but none of which the kids are ready for). Something slapstick-y maybe, without the sexual bits. I remember liking The Money Pit but don’t remember if it qualifies… anyone see it recently?
And the whole “too old” thing is weird. Sometimes it’s negotiable. They did really like Back the Future, for example. And Home Alone and The Parent Trap. But they won’t even make it through the trailer for Spy Kids… so it’s difficult to predict.
I guess these are old, but if, as you say, they sometimes can be persuaded, then:
Castaway
Stand By Me
Breaking Away
Popeye The Sailor Meets Ali Baba’s Forty Thieves
I recently caught Silver Streak (PG, 1976) and, other than Gene Wilder donning brownface to hide from cops, it holds up. I saw it during its original theatrical run – the same age as your youngest – and what few adult jokes there are went over my head.
Can be watched on Amazon Prime.
Going Postal From the Terry Pratchett novel. If only for the CGI of the “tidal wave” of letters.
I know it’s earlier than the time limit, but try “It’s a Mad, Mad, Mad, Mad World”. Stunning visuals in widescreen.
A lot of good replies so far…
I’ll add in a few “family movies” I’ve watched with my similar aged kids that fit the criteria of the OP, all animated, but eminently adult enjoyable. (I feel that the medium of animation is often quite underappreciated in this country as a serious one, but it so happens all of these are “family friendly”.)
Big Hero 6 (2014). A Disney movie, so on Disney+ which it seems like you’ve got, and Marvel derived enough for there to be a Stan Lee cameo (in the aftercredits).
It was fun even though it was predictably predictable to an adult (including the SURPRISE TWIST!), yet the ending was still surprisingly emotional enough to bring legit tears to my eyes. Baymax went from robot to friend to… parent.
Song of the Sea (2014). I think this may still be on Netflix, I know it was for a long time. A beautifully animated film based on Celtic fairy tales; it even has both an Irish and Scottish Gaelic language version, if for some reason you would be interested in that, though I don’t think Netflix supported them (you’d have to get the Blu-Ray or something).
The animation style is distinct from “Japanese anime” and the current “Pixar/Disney” styles, too, so aside from the storyline, very cool to see this
Kubo and the Two Strings (2016). Another distinctively animated film, with a fairly different story line from what you usually see.
Corpse Bride (2005). OK this one is a bit older than the 10 year window given in the OP, but if everybody enjoyed The Nightmare Before Christmas (and of course you did), and also saw/liked Coco, this is an interesting Middle Way from Tim Burton about love, and other things, reaching across the so-called barrier that is death.
Then there are a few “short animated series” (not “shows” like, say, Animaniacs, but ten episode or so story arcs) that are worth watching for any adult as well:
Over The Garden Wall. This one’s from Cartoon Network originally, so it’s included with HBO Max now. Truly magical. You’re thrown “in medias res” into two children apparently lost in some mysterious woods, and weird and magical things happen there, but then things are Gradually Explained And Explored. With sometimes hauntingly memorable (or just plain fun and catchy) music!
Infinity Train. This also originated on CN for S1 and S2, but is now on HBO Max (for S3 and the upcoming S4). Each season is separate, with a different “main character”, though there is some continuity from season to season. Again, a kind of “in medias res” beginning about a train that seems to go on and on, with each car having some kind of mini-quest going on (not always obvious why or what) to get to the next car, but why? And how did anybody get on this train?
Oh, and if you haven’t already watched the completed series (two seasons) of Gravity Falls on Disney+, why haven’t you? DO IT NOW!
(And finally, I totally concur: anyone who did not like The Iron Giant needs to search the couch cushion deep and hard for their soul that clearly leaked out during the opening credits.)
The Adventures of Baron Munchausen
I don’t see any of the Madagascar movies mentioned. Animated and hilarious.
Pity they ruled out Wallace and Gromit movies. Curse of the Wererabbit is one of the funniest movies I’ve ever seen. 
Stand By Me and Breaking Away are great, but perhaps better for high school or college age kids.
Assuming that the adults and the kids probably veto en masse, perhaps you could try a different method. On Fridays, the children pick 3 movies. The adults choose 1 of those to watch. On Saturdays, switch the pickers and choosers. It minimizes regret somewhat, while still allowing good movies to get past the veto stage.
Someone mentioned It’s a Mad, Mad, Mad, Mad, Mad World. It was too uneven (and “oldie”) for my kids. But they LOVED the remake: Rat Race.
Which reminds me, when they were little (8 and 11, and for the next five years), we checked movies out of the library. The. Same. Movies. Over. And over… so I saw Rat Race a hundred times. And The Rocketeer two hundred. Both great fun! Oh, and Hook! And Ratatouille, and Napoleon Dynamite. Luckily, I loved these as much as the kids did, because every line is burned into my brain…
And a lot of other “broad” comedies: Monty Python and the Holy Grail, National Lampoon’s Vacation, and The Naked Gun. I found episodes of Police Squad for them, too.
Those kids grew up before Sonic or Onward or Big Hero 6 or Zootopia came out, but I didn’t…
Oh, don’t forget TV shows.
We were surprised by how much our kids loved comedies that we didn’t think they could relate to: “How can you love The Office? You know nothing about working in one!” “Well, we do now… now shhh, next episode’s starting!” They also got hooked on Seinfeld and Friends. And of course The Simpsons.
There’s way more than that. The redshirts thing. The Captain always having to take his shirt off. Indeed the whole insane fan convention thing.
I watched it with my boys when they were about the same age as the OP’s kids and they loved it (“can you fashion some sort of rudimentary lathe?” has become a staple of our conversation) but there was a lot that I had to explain or which went over their heads.
Those are all asides, though; they aren’t major plot elements, except for the convention, which is hardly just limited to Star Trek. Setting the autodestruct is a staple trope of the series (even though they’ve only ever actually blown up one ship) and the only extended joke that requires that context to really get why it is funny. The film is quite accessible without reference to Star Trek largely because of the excellent cast and the balance of humor and sincere drama.
Stranger
I entirely agree with your final sentence.
But actually I think you viewing it through the lens of being familiar with Star Trek. I know the tropes a bit but I’ve never watched Star Trek. So the “00.01” thing didn’t seem that major to me because neither my kids nor I knew enough to know what we didn’t know. We just assumed it was the typical “down to the wire” thing that always happens in movies. It just sailed overhead unnoticed.
Contrastingly, the whole scene where Guy is terrified he’s going to die seems weirdly melodramatic unless you know what happens to redshirts.
I suppose, although the idea of a one dimensional, barely referenced character being killed off to demonstrate to the remaining characters how dangerous the situation is hardly limited to Star Trek; it’s another staple trope that can be found throughout science fiction and action, e.g. the killing of Ellis in Die Hard. When people first saw Alien it was shocking that Tom Skerritt’s character was killed off early, in part because he was a major character but also because Skerritt was basically the only recognizable actor in the cast; viewers were used to only apparently unimportant characters dying off.
The film certainly has a greater resonance if you ‘get’ all the references to Star Trek, like the launch sequence and the obvious Shatner-esque arrogance of Tim Allen’s character, but I think kids would enjoy the movie just for what it is, a fun space romp with a fair bit of silliness and a few sincerely dramatic moments,
Stranger
We watched with my 10 yo a few weeks back for our lockdown movie night. She’s watched a bit of Star Trek:TOS with me and liked it OK (I started her with Trouble with Tribbles), but it’s not something she requests. Anyway, she definitely seemed to enjoy Galaxy Quest. She got that some of it was poking fun at Trek, but mostly she enjoyed it on its own merits.