Treasure-for-xp all the way. At least as the bulk of xp. Haven't found anything else that balances ease of accounting, motivation, and open-endedness about which strategies are incentivized. As a player, finding treasure that gives you xp feels sublime - you've connected with something primal urge, gotten useful a tool AND have advanced yourself. It's a nice default motivation, supplementing whatever else in a campaign - players wonder what to do: I know, find some gold!
Versus fighting monsters/obstacles - it's a much less narrow motivation. Sneaking, stealing, subterfuge, diplomacy all become viable xp-increasing strategies with out the DM having to decide what exactly counts as "defeating" an obstacle if not monster death. And there is no significant incentive to just grind through monsters with no further goal. (And, the accounting seems to be a little less tedious!)
Versus milestones - in an even sort of open ended campaign, these seem to converge to just DM Fiat in my experience, which basically means that as a player I just don't think about it, and more or less assume leveling will happen once every few sessions. Which is fine, but you've removed a motivational tool from the toolbox, something that encourages risk and reward. Advancement being player driven adds a nice texture to a player driven campaign.
I've tried giving setting/campaign or class specific milestones for xp - clerics can establish shrines, wizards can learn spells, players can weaken a faction or gain the support of a powerful patron, or design their own goals. In theory this sort of seems like a great way to work things in a setting/campaign where gold-for-xp is not thematically the best fit (I like
these ideas for an Arthurian campaign a lot), but in practice, I always seem to forget about these as a player or DM. It's hard to keep a list of various achievements for various xps for in your head. Or players remember one milestone that seems the easiest and go for that, orthogonal to other campaign incentives, so it's easy to accidentally warp incentives in a gamey-feeling way. It can probably be very compellingly tailored in a particular campaign, but being campaign-particular, it takes some thought and design. Gold-for-xp seems to work out of the box, as long as you follow the general principle that greater treasure tends to be guarded by greater risk.
Versus social encounters - how do you determine whether and how much a social encounter was been won? If only there were some universal numerical measure exchangeable with in-world power and influence that we could attach to the downstream consequences of maneuvering a social or political situation in ones favor! (Being a bit glib here, this is maybe tough to get right, but broadly, if you have an incentive to get gold, doing better in all sorts of social situations can aid that. Monsters and patrons might know where treasure is, can help with logistics/transport/allies, and in general can make you more influential which means more opportunities to overcome enemies and get more gold).
In general, I think gold-for-xp is a little less limiting a motivation than maybe it first seems. It's not just for grubby sword and sorcery antiheros - I think it can work very well in a heroic campaign. Money and the power it can buy are the natural fuel for the the engine that fights against chaos and evil. It can purchase, upgrade and repair equipment, allow for the learning and casting of powerful spells, pay for training (if that is a mechanic you are using), buy ships, construct strongholds, hire hirelings and men at arms, pay for sage advice and knowledge, fund magical research and crafting, sway powerful figures, etc. It's a broadly useful (not universal, that would be boring) currency in whatever overall campaign goals exist, so it can subsume a lot of subgoals you might establish as milestones (e.g. a Cleric can establish a shrine using ....treasure!) . And when you acquire it, you are usually taking that power out of the hands or realms of the forces of chaos - we're not gonna reward stealing from the villager down the street with xp. Not crazy in this case for treasure to be a measure of achievement and power for good.