I’m going to second this. I’ve seen some episodes recently and while it was full of silly bits, there was a serious core in every episode that this was a team that was out to battle Nazis. It was a sort of Mission Impossible with jokes.
The first season or so of McHale’s Navy was like that, too, but it quickly moved to McHale saving the crew from their wacky hijinks.
ThelmaLou, apparently.
Mystery Science Theater 3000 - A janitor and two homemade robots watch terrible movies and talk over them. That’s ‘internet meme’ level of randumb elements, even before gilding the lily with the space station and mad scientists. But, it works.
But only after being endorsed by Dopers. Heck, I don’t buy toilet paper unless it’s endorsed by Dopers.
I came to offer this.
So, this one probably won’t fly, but Nova in the '70s & '80s. “Let’s get sciency-types [‘nerd’ was not in use then] to wax enthusiastic in overly technical language about subjects people don’t understand, and then over simplify it for people.” Everyone I knew loved it, watching the enthusiastic scientists with bated breath until the host or voiceover told them what that all meant.
The Blues Brothers movie is a fine example of throwing everything at the wall and creating a masterpiece.
*Lucifer *- the Devil gets tired of being in Hell, so he moves to Los Angeles to fight crime!
This is one of my favorite shows, and almost all of it is because of Tom Ellis, who pulls it off.
They know how silly their premise is, too. They even wrote lyrics to their theme-song riff:
“Crime-solving Devil
It makes sense.
Don’t overthink it.” 
I couldn’t stand the show, because I could not imagine ever finding WWII funny. But I found out that many of the actors were Jewish, and many of the actors had fled, survived, or escaped the Nazis. They were able to make the Germans the butt of every joke, and the good guys won every time at the insistence of Werner Klemperer (Klink).
Idea for a new thread: is there any better way to demoralize an enemy than ridiculing them?
Add MASH* to the list of war comedies. Started in 1972 with obvious references to the then long running Vietnam War.
Didn’t look to many that it would work. Became one of the most successful sitcoms of all time. The finale set Nielsen records.
FTFY version;
The Blues Brothers movie is a fine example of throwing a brick of cocaine at the wall and creating a masterpiece.
My late husband, who was a Medivac (Dustoff) pilot in VietNam in 1966-67, said that MASH captured perfectly the feeling, with all the conflicting emotions, of being there.
Warehouse 13. It’s kind of like a USA Up All Night version of the X Files, but charming, gung-ho actors with great chemistry make it work. Also, there are a surprising number of cameos from Star Trek actors.
**Psych **- especially as it aired against the ‘serious’ version of ‘The Mentalist’.
I’m still amazed at how fantastically good 3rd Rock is. I regularly rewatch it whenever I can.
Having just watched the concert version, I would add:
Jesus Christ Superstar: Let’s release a concept album by two unknown very young British menle about the last days of Jesus’s life from Judas’s perception. A very hard rocking concept album.That should really do well.
And 50 years later, we are still listening to this classic work
Is that like Homeland and In Treatment — a show based on an Israeli one? 
To quote a statement made by Andrew Lloyd Webber in this month’s Playbille magazine: if you were talking to an investor and said "I’m intending to do a musical about a founding father of America that nobody’s ever heard of and it’s going to be in hip-hop, they would probably say “Would you please go and kindly jump out the window.”
Yet the West End production of Hamilton just picked up 7 Oliver Awards, including Best Musical.
Though, to be fair, Psych did it first (it premiered two years before The Mentalist).