The only downside I see with Prestige Feats like this is that you have to wait up to four levels to take them. A prestige class could be entered with your next level if you qualify. I consider that a minor issue, though.
Yes, but it isn't a minor issue IMO: four levels is a
long time in our games. Even if you play every weekend, it usually takes at least a few sessions to level up, then multiply by four... Speeding up XP and level advancement just because one player needs to get to the next feat would be ridiculous.
That's the principal benefit from using prestige classes: you can design them as a short number of consecutive levels.
(This doesn't mean that prestige classes should replace feats! Just that feats cannot cover all possibilities)
I really like the idea of prc's being replaced by new abilities that are like innate magic items. They are a clean, clear bonus you get exactly when and if the DM says you do. Like all magic items in 5e, you aren't entitled to them, you get them when the DM decided it is appropriate.
No, this achieves very different results, and
adds on top of the character's power just like magic items. It can be a totally good idea on its own, but then the fundamental limit of this is that you need to do something similar to all PCs. So if one PC joins a prestige group and starts getting boons, eventually after a while you
must either pushes the others to do the same or compensate them with more magic items, except that you can't force PCs to share magic items the way you want.
Prestige classes, subclasses and feats work
within the structure of class progression, not "on top", therefore they are inherently balanced (unless of course you design benefits that aren't balanced in the first place).
The reason it failed is because they did not learn from 2e. Roleplaying requirements for classes or anything mechanical just doesn't work for balancing a it.
We need to think out-of-the-box and consider removing mechanical prerequisites completely.
All that matters mechanically is that the costs and benefits balance each other. The idea with prestige classes is to design a bunch of consecutive levels that would replace (or more properly displace) your base class levels, thus they should be worth the same as those levels. The difficulty is that level 3 is not worth the same as level 15, because higher-level features typically are more powerful. At least 5e doesn't have "dead levels" and has better multiclassing rules, and these should help. But why complicating design even further by introducing mechanical requirements which then needs to be taken into account as additional costs, and thus have prestige classes with higher costs that then needs better features?
Edit:
Because of this, the only truly useful mechanical requirement for prestige classes is LEVEL.
For some silly reasons, the 3e designers got into thinking that "level" was an inelegant requirement, and started a trend of never using it explicitly, but the truth is that they were always using it implicitly, just covering up with minumum BAB, ST or skill ranks. This trend was so powerful, that whenever 3rd-party designers dared publish a prestige class with explicit level requirement, they got flamed and labelled as "bad design".
In retrospective, this was really stupid and counterproductive! Character level is immediately clear, and simplify designs. You can then comfortably design a prestige class that immediately (at its 1st level) grants e.g. the ability to cast a spell of 5th level, by simply requiring minimum character level 9 to start taking levels in such class.
You can also call it
suggested required level, when its features are not so clearly tied to level, to let the DM have an idea but still decide freely.