To me, Milestone Leveling feels too much like DM fiat. The DM decides on a whim, when the Players Level up, and it usually feels arbitrary, even when I as the DM do it. Also, Milestone-Leveling takes away one way for the players to track their progress. XP is one of the two main mechanics in DND 5e to show progress (the other one being the Character Levels themselves).
XP is usually as much DM fiat as milestone level; after all, who decides when the PCs get XP, and what for?
And the feeling of progress is an important part of D&D and most games. If you don't have a feel of progress, if you feel stuck, if everything feels the same old, be it a game or real life, it sucks.
Like for example, in my game I DM with milestones (that started 4,5 Years ago), my players were not pursuing the main-goals of the story arc but some side quests. By my design, they didn't reach the milestones for a level-up for a long time and the players didn't like it, because they were stuck at that level. I could have changed the milestones, of course, but then I would have to change my whole milestone leveling system ...
Whatever happened to achieving personal goals? Way back there was this computer game called Myth: The Fallen Lords, and I decided I was going to finish the game with zero casualties. It was a personal goal, and I replayed every single level until I beat it without losing any soldiers. It was incredibly satisfying to figure that out for each level, but the game sure didn't reward me in any mechanical way for doing it.
Personal goals don't require a singular stat, be that XP or levels (which are just arbitrary XP thresholds that concrete advancements are hung on; that is, the two are actually the same scale). Clearing a dungeon is a thing you can track as progress. Rooting out a cult or secret cabal, member by member, is a thing you can track as progress. Collecting the parts of the tropey broken artifact is a thing you can track as progress. Keep a list of these, if the players need a number they can see growing.
Why couldn't you listen for, or ask, what your players' personal goals are? Then, when they achieve those goals, you give them XP, or a level-up. Yes, you would have to change (or ditch) your whole milestone-leveling system, but since that feels too much like DM fiat to you anyhow, maybe everybody wins.
Why couldn't you listen for signs of players being bored with the level they're at, and use that as an opportunity to give them a "milestone" level-up? If your players (or you!) are not having fun,
something needs to change. It might not be power level—but it might. Or, you could just make it clear that they players can explicitly state when they are ready for, or want, a level-up. This could be as a group, or individually (although it's likely that as soon as one person opts to level up, the others will follow in short order). Don't button down the details of your encounters too far in advance if you take these approaches, of course.
When you take out XP you will have to replace it with another way to track the progress to the next level, or it will feel arbitrary, and the players could feel dissatisfied and "betrayed" by the DM, when they think, they are stuck at a level.
XP is like a progress bar that the players can see. Watching a progress bar fill is a satisfying thing for most players, that's why a lot of games use those (and even Books and Movies - come on, everybody loved it, when Neville Longbottom got 10 Points for Gryffindor to beat Slytherin in the house cup in the First Harry Potter Book!). It is an easy way to bind the players closer to the game.
Again, XP is just one progress bar. And points for Gryffindor is a lot like faction status; most games that do that have many factions, even if a player is focusing on one at a given time. In my current Blades in the Dark campaign, we have lots of factions, and if our status with one goes up, our status with others is likely to change, too (in either direction). Some factions are allied; some are rivals. World of Warcraft, back when I played it, had similiar friendly and rival factions. That makes progress much more interesting and dynamic than a single bar that fills up. How might you fit something else like that into a game, where progress on one axis affects progress on another axis?
Again, there are plenty of ways to track progress, and again, it doesn't have to be to a next level (of power, or complexity).
If you remove it, the only thing that is left in D&D in a mechanical way to track progress are the levels themselves (there is also Wealth-progress and gear-progress in D&D 5e, but that is not very robust)
All other progress would be Ingame/"Story"-progress (we have to fulfill this quest, go in this dungeon ... ).
The game element of D&D—and any roleplaying game—is important, as is the "story" bit. But they can be important
in relation to one another, and not just for their own sakes (not that there's anything wrong with that!).
(Statement above revised based on private feedback.)
And also, yes, you can incentivize player behavior by choosing what you give XP for. That's a big bonus, too.
Again, that sounds like the DM fiat/whim you expressed a problem with at the beginning of your post.
For example, if your players tend to do the 1-Encounter-Day (doing one combat, then taking a long rest) when the dungeon delve, you can disincentivize that, by repopulating the Dungeon and only giving 1/10 of the XP for the Monsters that came back, because the players waiting a day to get their long rest (giving 1/10 XP feels worse than giving no XP). You can decide to not give XP for Murderhoboing innocent people and fighting the city guard after they want to arrest you for murderhoboing. XP is a great tool for DMs and Game Designers.
If your players enjoy 1-Encounter-Days, and
don't enjoy being punished for it, well, why are you making what should be an enjoyable pastime
not enjoyable? Maybe find ways to
incentivize rather than
dis-incentivize. Maybe talk to your players and find out what playstyle they enjoy most, both before beginning and as you play (since they may not know when you begin, or their preferences may change through play), adjusting whatever measures the group uses to track the various kinds of progress. Maybe observe the incentives and disincentives baked into the system and talk about those, and tune them to fit the whole group.
Will the game work without XP? Of course. Like I said, I DM without XP for the last 4 1/2 years, and it is mostly fine. But in my next campaign, when this is over, I'm bringing XP back, because it helps with the feel of progress in the game, it feels fairer for the players, and it gives me as the DM a tool to incentivize good player behavior.
XP may feel fairer, but it's every bit as arbitrary as DM-fiat milestones. (But I do see you've mentioned incentivizing after all!)