I have been using milestone XP for several years now, which (I feel) has at least partially contributed to apathy from several players. (Why bother making an effort to come if you get all the character rewards anyway?) Also, it's made players want to rush through content, avoid side quests and exploration, roleplaying encounters, and wandering monsters.
In a game like D&D, especially one as DM-centric as 5e, there's more to player rewards than just XP and advancing along the predestined progression of your class or build.
You character can be taken off those rails, for good or ill, at any time, by the DM, something as simple as an item too good to pass up - or cursed that you can't get rid of - or the campaign incentivizing different choices than you originally intended, emphasizing one pillar or type of challenge or enemy more than expected for instance. Then there's resolving actions, in 5e the DM signs off on what an action does, whether it succeeds, fails, or needs a check, essentially every time. If a player is engaged and pays attention, he's more likely to come up with actions that the DM will narrate success on.
The concern is that I'm getting ready to run an AP, and I don't want characters to have to level grind to get to the appropriate level for the game.
You can do both. You can start the game at a given level that fits the story, and provide enough opportunities for advancement to hit the right level at the right time (or a little before, or a bit late, depending on how the players behave). You can even have a hiatus between chapters in which everyone, through 'other adventures' or 'special training' or whatever jump forward to the next level you want to run. But, in the course of a given story, XP can be per session, instead of milestone.
You can win an RPG?
You can certainly lose.
Seriously, a TPK is a loss, no?
In a cooperative game, you can all win or all lose, and as long as you each contributed meaningfully, you each share in it.
I wouldn’t come back as being a trained monkey isn’t my shtick.
Better training monkeys than herding cats.