Make an Executive Decision and lay some Kurosawa on their asses! 
Kagemusha is something everyone should see.
Or should I use this one? 
It fails one of your parameters because it’s from the 1950s but I can’t recommend this movie enough for family watching.
The Million Pound Note starring the great Gregory Peck. A funny yet deep meaning film based on Mark Twain’s short story about how far wealth, or the mere symbol of wealth, changes hearts and minds.
Sorry, I didn’t see that there was an 8 y.o. in the mix. I thought it was teen/pre-teen kids only. “Tremors” may be a little too scary for someone that young, but it is as much (or more) a comedy than horror.
(I don’t get kids and scary movies. My 6 y.o. grandson has seen multiple Jurassic Park movies and other scary stuff that didn’t seem to bother him, but was too scared to watch “Beetlejuice” when we turned it on one night.)
Explorers clip: Robert Picardo as “Wak”, his main role in the movie.
I was hoping to find his introduction scene: “A horse is a horse, of course, of course…” Imagine making First Contact and the alien starts belting out that!
I wanted to show Time Bandits to my brother’s kids but he thought they’d have trouble with the accents.
And if you want to go old-school, I think the 1938 Robin Hood film starring Errol Flynn is still a great, fun film.
Galaxy Quest is an obvious homage to and satire of the tropes of Star Trek, but I don’t think you really have to be versed in the lore of that show to appreciate the humor. The essential joke is that the crew of the fictional show is presented with the opportunity to play their roles even though they are completely unversed in technology, diplomacy, or space warfare, and the humor primarily comes from that rather than specific references to Star Trek. The joke that most relies on familiarity with the show is the self-destruct device that always stops at “00:01”.
The face-melting scene in Raiders is obviously a melting wax head; it didn’t scar me in the way that watching The Thing the next winter did. There are some adult themes and violence in Raiders to be sure, but it is fundamentally a high budget version of cliffhanger serials that were made for kids.
Well, I did. I believe it was the first movie I saw Jimmy Stewart in (this was before It’s A Wonderful Life entered the public domain and became the Christmas movie despite the awful lessons it teaches) and he, Thelma Ritter, and Grace Kelly have an entrancing dynamic. It does start slow but once Stewart’s character starts suspecting Raymond Burr of killing his wife it moves quickly. And despite some of the deep subject matter (in addition to murder, there is attempted rape and suicide, the off-screen killing of a beloved pet, et cetera) it is definitely a ‘Fifties Hays Code film where all of that is subtle enough to not be obvious to a young child.
Stranger
I should mention that his “space pirate” looks almost exactly the same but uses a much lower camera angle, making him appear much bigger than he really is.
Think how you would be if your 12 & 14 year old took your car for a spin.
I don’t really watch newish movies for kids, so I’m sure my suggestions below will be easily vetoed.
King Kong (1933)
Miracle on 34th Street (1947)
Mighty Joe Young (1949 & 1998)
Twenty Thousand Leagues Under the Sea (1954) - My parents took me (and my misgivings) to see this when I was maybe 10-11. I was totally entertained the whole time and thought it was great.
Batman (1966)
Yellow Submarine (1968)
Close Encounters of the Third Kind (1977)
Superman (1978)
The Black Stallion (1979) - I was in my late teens when this started very late one night on cable. I loathe horse movies and I figured I’d watch a few minutes before I got sick of it. I ended up watching the whole thing and couldn’t believe I liked it.
Pee-Wee’s Big Adventure (1985)
Beetlejuice (1988)
Willow (1988)
Alternatively, you might consider something long and boring that will put the kids to sleep so you and the wife can watch something good.
I don’t think this goes far enough; consider trading whoever vetoed it to another family.
Perhaps you could trick your kids into watching older stuff with Mystery Science Theater 3000.
The original Muppet Show was only half an hour per episode. Maybe it could hold their interest long enough to get them hooked. I have fonder memories of the TV show, than of any of the movies.
Princess Mononoke is WAY too scary for kids. I won’t let my 12yo watch it yet, much less my 8yo. It’s got some straight-up body-horror sequences. It’s also my favorite Miyazaki.
Forgive me if this is an asshole thing to say, but have you considered not giving the kids veto power? It sounds like they’re getting to make this unnecessarily difficult. In my family, we take rough turns choosing movies, and as long as they’re appropriate, folks only get veto power over whether they themselves stay and watch the movie or go do something else. It lets kids see things they might not otherwise choose. And same for me: I end up watching some total dreck, but also some stuff that is pretty good that I wouldn’t otherwise have chosen.
This was one of my favorite movies as a kid. Recently tried to get my 9 YO son to watch it and it was meh for him. YMMV.
What about Goonies? That should hit the mark. Some “scary” parts, but if they watched LOTR I can’t believe would be too much.
Maybe the Jurassic Park movies. My 9 YO enjoyed them and wasn’t scared.
My brother said he watched Goonies as an adult, and it was great as long as you could see past the grotesque racism and sexism and fat-shaming. We gave it a pass.
Maybe it’s the nostalgia, but I still think Goonies was a great movie. (What sexism, by the way?)
Maybe expand your genres. I can always watch Apollo 13, but I’m getting bloody sick of Harry Potter movies, Star Wars movies, superhero movies, ‘sword & sandal’ movies… Stuff that The Missus likes to watch over and over again instead of new episodes of TV shows (like The Walking Dead) that are stacking up.
Anyway, last year we watched Ernesto’s Manifesto. Is it Science Fiction/Fantasy? No. But it’s new (2020), and it’s a comedy. Being sometimes involved in filmmaking, I liked the stereotypical ‘independent filmmaker’ story. Maybe it was the result of being shut in during the first COVID lockdown, but we both liked it and I had several good belly laughs.
ETA: Ernesto’s Manifesto VOD review – Entertainment Focus (entertainment-focus.com)
ETA 2: It’s on Amazon Prime.
I haven’t seen it in decades. Maybe he said able-bodiedness or something, honestly, I can’t remember; I just remember that he suggested a lot of the humor relied on gross stereotypes, and decided to give it a miss.
OK, I guess I can see the criticism. But honestly, a lot of media from prior to last year is going to be problematic if one looks at it from a present-day perspective.
100% true.
The White Seal and Rikki-Tikki-Tavi. Two Chuck Jones cartoons based on Rudyard Kipling stories.
Rikki-Tikki-Tavi is about a conflict between a mongoose and a cobra, and modern critics might consider it too intense for small children, but I watched it when I was little, and I turned out OK.
Chuck Jones also did an adaptation of The Phantom Tollbooth. I have not seen it, but it might be worth a try.
Considering what you’ve said about the kids, stick with the 2017 one. While there are a few call outs to the previous film in the 2017 version, it’s all but blink and you miss it, nothing is lost. The modern one is much faster paced, with more gags, and far less drama and unspoken tragedy of the original.
We enjoyed the 2017 one.