Yes, and they’re awesome!
The theory I’ve heard is that they are all session musicians under contract with different record labels, so the only way they can play together is incognito. The costumes and double entendre lyrics are funny for a minute or two; after that it’s just really solid funk.
Gimmick bands don’t have to be bad. The Blues Brothers sold because they put together one of the best rhythm and blues bands ever, and sang classic songs. Belushi developed into a good singer, and Ayckroyd could play the harp pretty well. And they were in the tradition of the genre - Cab Calloway had a gimmick or two himself.
Wasn’t David Bowie’s career one long gimmick? Ziggy Stardust, The Thin White Duke… He was still great, but he wasn’t just about the music.
Does ZZ Top fit? Alice Cooper?
Some of these are “High Concept” bands: you can describe, in one sentence, the “premise” of the band.
The OP’s Okilly Dokilly is an example of this, as is (the first example I thought of) Dread Zeppelin. I’ve known about them for a long time and could tell you what their shtick is, even though I don’t recall ever seeing or hearing them perform. Other examples that I don’t think have been mentioned yet: the Rock Bottom Remainders, and ApologetiX.
This category includes, but is not limited to, “cover bands with a twist.”
Another category is Fictional Bands—bands that were intentionally created for a movie or TV show, though they may have taken on a life of their own. This would include Spinal Tap, The Rutles, The Monkees, The Archies, The Blues Brothers, and Flight of the Conchords.
The Pipettes was a trio of women singers doing a great version of the classic “girl band” shtick. Their 2006 album is great. But nonstop lineup changes, etc. meant that was pretty much it.
It’s interesting to hear 60s style songs about modern topics.
In the early '90s, Green Jellö billed themselves as “the world’s first video-only band.” They released a VHS tape, rather than a traditional album, consisting of 11 videos (they later released a “soundtrack” on CD). “Three Little Pigs” got heavy rotation on MTV for a little while.
Mummified (and top hatted) Torontonian Nash the Slash
Hate much? 
I say Rush. Their gimmick is to convince people that they don’t suck, even though they do.
Surf guitar music made by men wearing Mexican wrestling masks.
Not the last one. Brett and Jemain were performing together as FotC for 6 years before they got their radio show, and 9 years before getting their TV series.
As to the OP, I would say that the difference between a band (or performer) with a gimmick, and a ‘gimmick band’, is how deep the gimmick goes.
A ‘band with a gimmick’, it’s a fairly cosmetic thing. Kiss is a band with a gimmick. The makeup has ‘characters’ associated with it, but their stage personae don’t really follow through on the characters (ok, Gene’s does a bit), and they don’t inform the lyrics, or the style (and given the characters range from The Demon, to The Spaceman, To the Catman, how could they?). Simply being weird wouldn’t count, unless they’re weird in a very specific, very consistent, very intentional way.
A ‘gimmick band’ would be one where the gimmick goes all the way through. GWAR is a gimmick band. Their monstrous personae aren’t just costumes, they’re full characters, and their music, both style and lyrics, are informed by the characters. I’d place Rob Zombie at the lower edge of someone I’d comfortably call a ‘gimmick performer’.
‘Fictional bands’, I’d say are a different category, unless they have some gimmick beyond that.
Some from my music collection that I haven’t seen mentioned in the thread so far, in each category…
Performers with a gimmick:
Tomoko Kawase’s solo projects tommy February6 and tommy Heavenly6…while there is some solid (if not entirely internally consistent) conceptualizing to the two, aside from the most recent Halloween EP, which crossed over the two in the lyrics, not just the video (I’m actually not sure if that one had a video associated with it), it’s primarily cosmetic.
Gorillaz - once again, there’s some pretty solid lore, here, but it only follows into the music in a very surface way…and the cartoon element is a bigger part of the gimmick than the actual histories, relationships, and personalities of the characters.
‘Gimmick bands’:
Lordi - monsters and horror!
Ghost - Satan and horror!
Steam Powered Giraffe - adorable steampunk automata! … Combo breaker!
Both Lordi and Ghost used to have kayfabe, as well, refusing to do band related public stuff out of character. (Ghost stopped after a lawsuit by some former Ghouls resulted in the identity of the lead singer (and only permanent member) coming out. Pretty sure Lordi still does, though.)
SPG is as gimmicky as it gets, and they get double bonus points for having their gimmick extend to their costumed handlers / merch table folks.
I can’t believe I’m gonna be the first person in this thread to mention The Aquabats and Peelander-Z, two bands that combine music with themes from kids’ action TV. The Aquabats are pretty famous and well-respected enough that they’ve had an actual TV show (and a reboot / continuation is currently in the works). Peelander-Z is less famous, but still has a significant cult following and a very good documentary on Netflix.
I’m with you. The Stones did just fine without the Beatles comparisons. It’s funny, The Stones had plenty of fans such as The Beatles, The Who, Clapton, etc…yet people still have this idea that there were weak tea. Nonsense. They have an outstanding legacy and are well established as some of the best rock writers of all time. The Stones grew in strength as the Beatles slowed to a stop. I like both bands very much but I admire the Stones a great deal for more or less inventing blues/rock. As the 60’s passed they kept growing and putting out better stuff than ever.
Just my .02
Not to mention Dolly Parton.
Of course you mean Paul Revere and the Raiders, who even wore funny costumes.
Or did you mean Jimi Hendrix?
Have all-female rock bands gone from being gimmick bands to bands with a gimmick? Up through the 80s, I feel like bands such as the Go-Gos or the Bangles weren’t take seriously on first blush. Today, a band like Sleater Kinney might get some extra notice for having an all-female lineup; but they’ll quickly be judged based on their own merits.
Me First and the Gimme Gimmes have been around for over two decades. They do punk-rock covers of songs from all over the musical spectrum (from City of New Orleans to Mandy to Uptown Girl to Rainbow Connection), and they’re pretty darn good.
Oh, that reminds me of The Iron Maidens. An all girl Iron Maiden cover band. They are great!
Insane Clown Posse are a gimmick band since the have been around forever, but I only know of their gimmicks and not one note of their music.
There is damn little that Mexican wrestling masks can’t make better.
Because they don’t have one note of ‘music’. 
I prefer to think of Weird Al as an Incarnation.
King Gizzard & the Wizard Lizard are both a band with a gimmick and a gimmick band, IMO.
Gimmick in name only - The Wailin’ Jennys