I feel like this may be more true of competitive games....chess and tic-tac-toe being examples that were mentioned....but in a collaborative game such as most RPGs, I don't think it's quite as necessary as all that.
Fairness is certainly an important aspect of competitive games. Chess may not be perfectly balanced, and tic-tac-toe very badly so, because of the advantage of going first, but in a tournament you play multiple games, so each plays white some of the time, and it's reasonably fair.
In a cooperative game, fairness a bit less important, because you just need to be fair to the cooperative unit, as a whole (and, really, it doesn't even need to be /that/ fair, if you're 'playing against the game,' the game's not going to get it's feelings hurt - and, D&D is probably an example of that, as well, it tends to favor the players). But, balance becomes
even more important, because you're not trying to provide meaningful/viable choices to just one player, but to multiple players all trying to contribute to a single goal,
as individuals.
Of course, it all depends on what you mean by balance....there are a lot of ways we could apply the term to an RPG and to D&D in particular. But given the OP and the way this conversation has gone, I look at it as a game not having "clearly superior options" in the form of feats or class combinations.
Feats and classes (and combinations thereof) are choices, yes. It wouldn't be /just/ that, but they're examples the OP directly referenced, he's also indirectly looking at spell choice and such, too, I'm sure...
...as far as that goes, the question is whether those choices are 'balanced' - by my preferred definition, are they all meaningful & viable - and if they're not, are there a few OP culprits that can be eliminated or nerfed so they stop obviating other choices. My personal conclusion is that the inclusion of feats & MCing simply isn't worth it, but one could go through and selectively ban/nerf/buff various feats to address the issues Zapp & others have found, rather than just toss them all out as a unit...
Maybe one possible solution is to ban multiclassing outright?
Ding! Feats & Multi-classing are both explicitly optional in the PH, to begin with, you just decline to opt into them! It's not even as ringing a condemnation as 'banning,' you just choose not to deal with the added complexity, and the problem vanishes.
(To, Zapp would point out, be replaces by the problem of DPR weapon-users being overshadowed, but no solution is perfect...)
And I think that's clearly not the case. Sure, a feat like Great Weapon Master may be seen by some as a clearly superior option, and a player who doesn't select it is not being "optimal". But that's only when combat efficacy is the only goal.
Or if combat efficacy is a class's primary meaningful/viable contribution. Zapp isn't worried that a Druid with GWM is going to outshine a knife-throwing wizard, he's concerned with the Fighter with SS out-shining the TWFighter or the S&B fighter or what-ever, and, if he doesn't allow the feats, also with the more DPR-oriented classes like the fighter not shining sufficiently, at all.
It's a legitimate concern, but probably not one best addressed by extensively re-writing already-optional rules, IMHO.
This is the fundamental flaw with CapnZapp's premise; it's an issue for his game (and certainly some others', too), but that doesn't mean that the game needs to be revised.
Revising 5e isn't even really on the table, it's not designed or structured to be errata'd into a better game as it goes, it's designed as a common starting point that will be customized to varying degrees by each DM who uses it. Any revision of the game could pull the rug out from under some of those customization efforts - and they may well constitute a great deal of effort, indeed. Each revision also creates a schism in the experience of the game before and after the revision, not like edition schisms, obviously, but why risk it...
If you want to apply the term balance more loosely....perhaps as "most of the existing options will appeal to someone"....then sure, balance is required. Because the goal is fun, so as long as all the players are enjoying themselves, then the game is balanced.
Not s'much, no. Imbalanced games generally have a balanced sub-set, once the chaff is winnowed away. Such may or may not be enough to constitute a worthwhile game. A solved game, like tic-tac-toe, for instance, not really. A vastly more complex game like D&D, though, even when you have eliminated the 'trap' and 'chaff' options probably has a fair amount left. If it's enough for a given group, they can have fun with it.
If the imbalances aren't all winnowed away, then the 'system master,' who can leverage those imbalances is going to have just that much more fun, too, until everyone else has gotten wise... ;>
Maybe, in the end, though, it's only an aesthetic/value judgement that a game should even try to set as few traps and leave as little chaff on the floor as possible?