The benefits:
This system would allow more experienced players, who have had the extra time to understand their skills and core rotation to curate a more enjoyable experience in lower level dungeon based on their playstyle with the help of all the extra skills they've unlocked through their journey so far, that they wouldn't have had at the level of the lower level content. Whether it be prioritising extra weavable OGCDs, more utility, raid support mechanics, players can select their skillset based on what will help them enjoy content that honestly, is easy enough that it doesn't remotely depend on the current skillset of the level available to it.
It would also make the game more flexible and engaging at lower levels, providing a trickle of new skills controlled by the AP limit, allowing them to experiment with what matters to them and the players around them, skill by skill, until they eventually have access to and understand all the skills within that threshold. I genuinely believe it would foster a better, deeper understanding of the skillsets a player has, if they are able to make a wrong choice, be punished for it, and upon evaluating that wipe or that bad run, understand why another move was a better option.

The struggles:
As always, with any amount of player customization, there is the significant risk of the system being abused to ruin others' enjoyment of the game.
In this case, i feel like it comes at the risk of two things; one, potentially making old raid content significantly and unexpectedly easier [more than expected, at least]. I feel like this is also double-edged though in that it might foster extra engagement in old high-end synced content by creating this community of people with 90-level skills trying to run the most meta build they can.
...This, however, comes with the risk of equally fostering a community willing to punish people for making the 'wrong choices' with their skills, or worse, for joining lower level high-end content with only the skills unlocked up to that point.
There's also the issue of finding a solution to the obvious issue of 'what if people... don't set threshold skills at all?'. I feel like the solution isn't to lock-out duty finder until all AP has been set- It feels unnecessarily punishing for something that could be an accident.
Implementing a simple and digestible solution to picking what skills get synced out when you get synced down to a lower AP is difficult. Maybe it could be like Nier Automata's system in that the order of things placed into the AP bar matters when it shrinks, but that might still be on the complicated side for new players.
There's also the matter of balancing what skills are picked as static level unlocks and remain synced, and what skills are available to use in lower level dungeons. Too many, and it becomes too easy for it to become too punishing and too troll-able; too few, and the system may as well not exist, for all the difference it'll make.