It is, and then again isn't, a separate conversation; in that when considering changes to any class one can't just look at the single class in isolation but must also consider the effects of those changes when someone multiclasses into (or out of) the class you're changing.
No, it really doesn't.
No one was discussing the One DnD druid and asking how changing wildshape would cause problems with multi-classing. No one was talking about the Paladin and how their smite changes would cause problems with multi-classing. No one discusses the wizard in the context of multi-classing.
So when discussing Fighters, Barbarians and Monks, we don't need to discuss how they might cause problems with mutli-classing and hold off on improving those classes until after we've fixed multiclassing.
The only viable way to prevent - or at least tamp down to a dull roar - exploits and munchkin-ism is to bake in rules that prevent those combos from occurring.
You can't prevent exploits, because exploits are the point of the rules of the game. Unless you think that combining feats and class features is something that needs to be prevented.
This isn't a big deal, it has never been a very big deal, and it isn't worth leaving warriors to languish just to make yourself feel better that someone isn't playing the game wrong.
Here it depends on one's design focus and agenda. Are (the hypothetical) we designing to specifically encourage group or party play, or are we designing for solo play (as in, one player, one character, and a DM)?
This is a big divide; in that characters designed for group play can be specialists with clear strengths and baked-in weaknesses while characters designed to work well in solo play need to be able to more or less do everything.
Thus, characters designed for solo-play are likely to overall out-power characters of the same level designed for group play, meaning we either can't design for both at once or, if we do, there has to be some restriction on the solo-play characters if-when used in group play. My own answer here is to not even try to design for solo play and instead focus on group play.
However, players also come to recognize the bit I highlighted; and realize it's in their better interests to try to build those do-it-all characters. Here, the game either needs to push back at the design level or somewhat sacrifice its group-play design paradigm; as if one character can do it all then what's the point of the group?
Let us consider the artificer.
Can the artificer tank? Yes
Can the Artificer deal good weapon damage? Yes
Can the artificer heal? Yes
Can the artifcer stealth, explore, and deal with traps? Yes
Can the artificer use spells to control the battlefield? Yes
Cant he artificer use spells for utility? Yes
So, here we have a jack of all trades. All ready in the game. So what's the point of a group game with a group of players, if the artificer can already do everything?
Whatever answer you come up with, it probably applies to being able to multi-class a rogue and ranger, or a fighter and a druid.
Balancing them (against each other) is going to be difficult anyway. And yes, some would be damage-plus (e.g. full damage plus pushback), some would be damage-or (e.g. my knock-prone idea), and some could even be part-damage-plus (e.g. half-damage plus disarm).
The thing is though, I don't like your idea being purely damage-or, because a pure damage-or is going to only be situationally useful. Meanwhile damage-plus will ALWAYS be useful.
I talked about this before, but Prone isn't always useful. I play a barbarian in a game right now whose teammates are all ranged characters. Knocking an enemy prone is the single worse thing I could do to help my allies. "Well then don't take that ability" is all well and good to say as a response, but we all know having an ability that is situationally useful makes it less powerful than an ability that is ALWAYS useful.
I get this, and while there's some low-hanging fruit there it's not an easy fix overall.
First off, from their initial knight-with-extras roots Paladins have gone much too far towards being casters. As other caster classes have emerged, I'd have pushed Paladins more toward the knight side; even more so as there's no specific knight or cavalier class any more and that's a pretty major archetype.
And yet, the vast majority of people have no problem with the paladin design, and actually one of the bigger complaints about paladins is they don't use their spellcasting enough.
Bards as currently designed are superfluous; yet again the whole class needs rebuilding from the ground up. It shouldn't be able to replace a Rogue's role in a party. And both Bards and Wizards have become too resilient and Wizards have become, dare I say, too good at combat - their once-weaknesses have been filled in with no corresponding drawback added.
Monks, however, could do with a bit more to them; they're the obvious on-ramp for a bespoke psionics system but for some reason WotC don't seem to want to go there.
So, your solution to the problem shown of caster design being such that an all-caster party is viable... is to nerf them. Even though none of those classes are largely seen as being OP.
Ideally, your Bard-Pally-Wizard-Cleric party should find itself really lacking in the stealth/sneak/scouting department; but beyond that it's viable.
Sounds like someone has never heard of invisibility, find familiar, or dozens of other spells that can handle stealth and scouting.
That all said, yes, in any edition a Fighter-Monk-Barbarian-Rogue party would be seen as having holes in its lineup. Throw in a healer of any type, though, and it's quite viable.
So, add a spellcaster and an all warrior, non-caster party becomes viable. Why, that's brilliant! Who would have thought that the best way to make a party full of non-casters viable is just to add a caster to their line-up.
Part of that, I think, comes from the emergence of "social mechanics". Time was, you could for example viably play any character with half-decent Cha - even a Ranger - as a negotiator, thus giving it more to do than just bash things or sneak around. Not so much, now.
And this gets back to my point about characters who can do a bit of everything. Ideally each class has something it's the best at (or at least very good at) and has something it's the worst at (or very bad, anyway). Rangers should, for example, be the best at outdoors exploration while Rogues are the best at indoor exploration. Druids should be the best class to have when outdoors and the worst to have underground. And so on.
And yet "social mechanics" have been a thing for 23 years or more. Probably more. And people generally like them.
So, maybe, just maybe, instead of nerfing other classes to no longer be solid mid-tier choices for the majority of roles... we could buff pure martial characters to be solid mid-tier choices for more roles?
Again, you are fighting so hard to nerf instead of buff, because you think that you are fighting power creep, but you aren't. What you are fighting is to keep pure martial characters out in the cold, because as I already demonstrated, they aren't actually needed in a solidly built party. But they are popular, people like the fantasy of them, so why not let them be slightly better at things they currently cannot meaningfully contribute to?
Even if we want to continue having Warriors be useless in social situations and exploration situations, and only be good at combat, we can STILL improve that so that they are actually the best combat class by giving them solid battlefield control abilities. Because right now, they can't, battlefield control is essentially solely the realm of casters, except for a single feat combo (sentinel+PAM)