If 5e PCs can't buy magic items, what do they do with all their money?

The answer is "whatever they want" - and if they come up with interesting stuff to do it's good to award them XP - one reason individual XP works well in 5e, though group awards are also fine. Castles, ships, armies - armies are actually useful in 5e! - henchmen/retainers... a 15th level 5e Fighter may well want a pet NPC Mage and NPC Priest on his payroll, along with a garrison of men-at-arms for his castle. And the stats from the back of the 5e MM work fine for this.

I think it's ok in 5e to let PCs occasionally use their vast wealth to search out or commission magic items, but the GM can/should make them arbitrarily expensive, just as the GM should decide what price an NPC will offer for those Boots of Flying; make it an in-game event, not book-keeping.

The trick is to get away from the purely character-focused approach of 3e-4e. You no longer need money for your 'build'; money is a means of interacting with and shaping the world around you.