I mean, you could do that, but you're sort of pushing into story-based awards any time you do. In your hypothetical situation, a group could get no xp at all for engaging with all three factions, taking on all three assignments, and working on all three without actually completing any. One of the joys of sandbox play, to me, is all the dangling plot threads that pcs leave behind them everywhere they go. Milestone xp seems like it would force the pcs clean a lot of those up in order to gain xp.
But yes, you can use milestone xp in a sandbox. I just feel that the very concept of milestones results in a strong push toward more story-based play. And while there's nothing wrong with that, it's not my preference.
Milestone leveling removes xp from the game entirely and instead just awards levels. The systems are a bit different.
So, in older editions of the game, XP made a lot of sense. But this was because of a few features of those games:
1. Classes needed different amounts of XP to progress to the next level.
2. Characters with high ability scores earned more XP than those with low scores.
3. Each class had a few optional bonus ways to earn XP.
In addition, older games tended towards this idea that PCs didn't always adventure together - sometimes, three players would grab whatever PC they had on hand to go explore the DM's dungeon. It wasn't always the same five or six PCs adventuring together - the party composition was much more in flux.
With that in mind, it was pretty much a given that every PC had different amounts of XP, and there was a wider spread of levels in the adventuring party.
Compare that to modern D&D versions, and things are a bit different. Each class progresses at the same XP limits, there are no bonus methods to gain XP in the core game, no XP bonuses for ability scores, etc. Plus, the party is assumed to be static - it is, I'd guess, more common than not for every PC around the table to have the exact same XP total (or near enough).
I'd say that if every player at the table has the same XP amount written on his or her character sheet, experience points are really just an illusion. By having the DM say "at the end of this adventure, you all level" or something similar, it saves a bit of bookkeeping and really doesn't change much.
Plus, as it was said earlier, players do what they're incentivized to do. If you remove XP, you are incentivizing the players in a different manner. If you level PCs every six weeks of real play, for example, the characters can really do whatever the players most enjoy. If you level them up when they progress through the adventure, you're rewarding players for keeping the game moving.
It creates a different thing than "Man, I need 50 more XP. Who wants to go orc hunting?"
I've mostly done the XP thing in the past. Lately, I've made a tentative switch to milestone and found my group loves it. In my next campaign, starting next month, we'll be following a "1 level per adventure" format that I think will be fun.
One just has a layer of abstraction the other does not. Otherwise, they are equivalent. The only question for the GM is whether the ability to sub-divide the bookkeeping into smaller pieces is useful.
Actually, I feel pretty strongly that XP for fighting is the way that makes the most sense. Not that it's perfect, but I feel like it makes more sense than most of the alternatives, especially in 5E. The only real problem with it is that spellcasting also progresses through combat, and you'd think that spellcasting would be a broad enough ability that you don't need to fight in order to improve it - there should be some order of pacifist wizards out there, and they shouldn't be stuck at first level forever - but then I remember that the rules are only really meant to describe professional adventurers. Presumably, non-adventuring wizards can progress through study and practice, in the same way that NPC carpenters can improve their skills.I think @Saelorn was essentially just suggesting a way to explain how "XP for fighting" can be viewed to make it palatable if you think it works weird. He could probably just as readily offer a way to explain milestone XP if someone thought that was the system that worked weird.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.