(Was everyone in A-team ok in a fist fight? Did all of them have at least two important skills?)
Yes, though Templeton usually hurt his hand punching people, and being punched typically meant he had to get new crowns put on his teeth.
You can certainly point towards each member of the A-Team having their own specialties, and times when one was obviously better (B.A. was hands down the best in a physical brawl), but the analogy breaks down a little (though it's probably better than my car analogy, lol) when you remember the A-Team is a TV show, not a game.
As a DM, my first priority is to make sure everyone is having fun. People have fun in different ways, but if someone is lagging behind a bit behind the others in their contributions, I can tell they might get frustrated. When I ran 3.x and Pathfinder 1e, there were legitimately classes that did not perform as well as others most of the time, and their true moments to shine were less frequent.
I certainly tried to adjust my game to give everyone their moment, but it's hard, especially if what you're good at is, kind of boring. Take the archetypical dungeon crawling Rogue. Being good at finding/disabling traps is important, but there's not always a trap to find, and if there is, the whole process is reduced to "we move at half speed. Does the Rogue find the trap? Y? We take no damage. N? We take damage. Does the Rogue disable the trap? Y? We take no damage. N? We take damage."
Sure there are interesting traps that merely slow down the group or alert enemies, but you have to work to build a trap that is more than a binary state.
That's not enough of a specialty for a character to have to justify them being slightly worse at everything else in the game. I didn't think that when I played AD&D, and I don't think that now.
It's like the Ranger being the "wilderness guy". It's basically a bunch of ribbon features that pretty much say "yeah, I win exploration challenges". That's not enough to justify making the Ranger worse than other characters either.
Now, sure, a class shouldn't be able to do everything, and I wouldn't want it to be able to, but there's not a lot of upside in my opinion to saying "you're less good at combat",
IF combat is a disproportionately large part of the game.
Sure, you could build classes that are behind the curve because they are meant to do other things, and maybe put warning labels on those classes, as I believe Micah was suggesting, but why? Why do we have to have these extra classes laying around, when it would be easier to give everyone a defined combat specialty first, and then give everyone a secondary role?
I mean, put another way, what is the weakness of the Paladin? They are just as good as the Fighter until level 11 in raw fighting ability. They have on-demand burst damage which can be pretty high, the utility of being a half-caster, grant one of the best party buffs in the game, and have additional healing resources, and a ribbon feature that makes them good at ferreting out undead and fiends, and all of this without even taking their subclass into account! Their focus on Charisma means most Paladins will be viable social characters (not saying other classes can't be, but it's a role the class can naturally fall into)- the Paladin may be the perfect class from a design standpoint.
Now some people don't like the class identity, or feel that Paladins have extra baggage, but the class itself is solid and no longer even has a built-in "and the Paladin falls mechanic"; the PHB offers no penalty to breaking one's oath, and the Oathbreaker is an option in the DMG (and isn't a downgrade to the Paladin in any respect).
And I realize some people take issue with this, but the point is, the class as presented works, and works well, and has it's own definable strengths, and never falls behind. The Fighter may at higher levels put out more reliable damage over the course of the day, but even without resources, the Paladin offers a lot to their party members.