Starting with the PHB numbers is flawed. Because not everyone with a PHB would play using a Prestige Class (since there were no PrC in the PHB). You have to use the number of copies of the splatbook being sold, which could be as low as 50k.
Fine, each splatbook didn't have 1000 PrCs in it, though. What single supplement introduced the most PrCs? Assume it also sold the least copies, that's 50k chances for each PrC in that book to get played at least once. Doesn't seem that far fetched.
Isn't that just a big ol' variant of the Oberoni or Rule 0 Fallacy?
In a sense it is, just a variant that might not be so fallacious, afterall. It isn't a game that tried to be balanced, failed, and then hides behind the ability of the DM to fix it. It's a game that wasn't meant to be played a particular way, so didn't try to be balanced, but left that sort of tuning to the DM, upfront. Balancing it the way you like is just part of running it, of 'making it your own,' not a fix. Indeed, if another DM ran exactly the way you did, he might not find it 'fixed' at all, he might even find it 'broken.'
But, say, there was one such class? Which one was it? How do you figure that out before you even publish it, so you can exclude it? What's the gain of publishing one less PrC out of hundreds?
The rules of a game aren't flawed because they can be ignored, or one or more "house rules" can be made as exceptions.
It might be fair to say they're designed to a different standard. 5e rules aren't built to the standard of a balance that 4e rules were, nor to standard of RAW-coherence that 3.5 rules were, but neither 3.5 nor 4e were built to the standard of DM Empowerment that 5e is. Different design goals, different designs.
Or, as you're arguing, content doesn't need to be balanced because the DM can control the balance.
In essence, yes. When you design content with the idea that the user will only use the bits he wants, the way he wants, you don't have to worry so much about how each and every bit interacts with each and every other bit, or how every bit balances with every other bit.
I think it's safe for game designers to assume that DMs don't want to put out fires at their game table and work around problematic options.
No matter how time-hallowed and long-since-worked-around those options may be?
That DMs don't want to have to nerf options and make hard one-sided rulings to keep PCs in check.
The broader set of GMs might not want to, but DMs have been doing so for so long, we should mostly be OK with it, by now - even a little put out if we're not. Though, you don't have to formally house-rule or 'nerf' anything, you just have to keep the game varied, fun & interesting. A weapon that does 0.5 more DPR than another, or a class that lacks daily resources, or an obscure perk rom an obscure supplement, or whatever else, isn't that hard to deal with, among the set of things present in a single party of PCs, which is all the DM needs to deal with. That doesn't change appreciably if the players chose from a lot of content or a little.