Personally, I think prestige classes are unnecessary in 5E. That design space is adequately filled by full classes, subclasses, and the new, larger feats.
The prestige portion lends itself very well to feats with setting based requirements. Membership of a particular guild or order, for example. Subclasses fill in for prestige classes that are variations on a base class. Finally, concepts that are two big for feats should probably be fleshed out into full classes.
Well, the point of Prestige Classes have always seemed to me to be for two reasons:
1) To bring in new mechanics to the game that are powerful enough to begin with that you wouldn't want to start them at 1st level (thus them not being actually classes).
2) To have a "game produced and written" story or narrative evolution of your character as you become more powerful, rather than the player just stating for him/herself "My character is now a Drunken Master!" (or whatever new character narrative the Prestige Class describes.)
As far as the latter point is concerned... many examples of the "generically-named" PrCs have been subsumed by other systems. Do we need the game to spell out what a 'Justicar' is? Or a 'Blackguard'? Or a 'Duelist'? Not really. Those kind of PrCs that are really just alternate class names can more than be covered by the system as it is. We don't need the game to make us a prestige class to become a 'Dwarven Defender'... if we're a dwarf and we have the feat, we're pretty much already that if even we don't have four or five paragraphs written by WotC describing what it is.
And as far as the former point... the problem ends up being just what kinds of additional game mechanics need to be introduced into the game, and how much of that stuff do we really need? And how useful and/or balanced are those mechanics over the mechanics we could get by just taking another level of our class, or start multiclassing? I mean... if all we had was the Basic Game and there was a Barbarian prestige class that introduced a 'Rage' mechanic (which is something many people have hoped the game moved to in the past)... that would be the kind of thing we'd be looking for. But now, once you get past the mechanics found in the 12 classes we're going to have (plus all the sub-classes)... what other mechanics are there and do we really need them? If the answer is no... then the need for prestige classes has passed.
Truth be told... rather than prestige CLASSES... I think we'd be better off with prestige BACKGROUNDS. Backgrounds you take at a certain level that are based off of what you have done in the campaign up to that point-- maybe an organization you've joined, an effect you are under, a rank or job you have attained, something like that. And that's where we can have described in four or five paragraphs what being a member of 'The Harpers' means (for example), and have them give us proficiency in a couple new skills, plus a new background trait. That seems to me a little more useful, a little less book space intensive, a way to get your character more skills, and a way to avoid needing to balance new mechanics against all the mechanics already in the game.
