I see no reason what was once "prestige class" couldn't be done as feats.
I would recommend the following parameters:
1) Not a "chain" per se. But, like an uber Feat-in-three-parts. While yes, it could be done, up to 19th in 5 parts, but if you make it 3 then characters who might not take their first level of the Feat until later in their career can still become a "full" <whatever the Prestige Feat is> by 19th. I also feel that if the "PF" can't be boiled down into 3 parts, then it is not necessary/too complex. This also means, for those that start/grab it right away at 4th, you're gaining the final piece of your Prestige feat by 12th, leaving you with a theoretical near-1/2 of your character's career [assuming a 20th level cap] to have their full suite of special abilities. Seems to improve flexibility and can involve story things to lead to them all along their level progression instead of "well, ya missed the boat at 4th. Never gonna make it now."
2) I agree that it should be left to the normal Feat selections, with a stipulation they can't be taken before 4th (to avoid the variant humans). I further concur with the "in story/RPing" focus and stipulate playing/working your character toward these goals, not just "pick a feat."
3) Each 3-part Feat should have some fairly simple to achieve prereq's that are not necessarily tied to any single class. So as to avoid powergaminess/"gaming the system" shenanigans. The Arcane Archer, for example, could stipulate the character must a) Be proficient with pull bows and b) Has access to Arcane magic. This would mean nearly half (maybe more) standard PCs could conceivably become Arcane Archers and even more if, say, a Champion Fighter or Assassin Rogue takes their 4th [or 8th, 12th, etc...] level feat for Magic Initiate: Arcane. Then their next feat could be Arcane Archer. Other classes: Wizard/Warlcok/Sorcerer could use the 4th level feat - or 1st level variant humans to get their prof in a Long/short bow and go on from there. Eldritch Knight and Arcane Trickster, arguably Rangers [if you flavor their magic as Arcane], could dive in as soon as they're able to take a first feat...which kinda makes sense.
4) Taking the 1st part of the Prestige Feat does not necessitate taking further pieces with your future feat selections. So you could end up with "Arcane Archers" of varying levels of capabilities at various levels.
So, yeah. I'd go for something like that. A 3-part "Prestige Feat" with 2 prereq buy-in seems fair and flexible.
[EDIT: and no Multiclassing required.

/EDIT]