Unarguably, they don't need to do that. They want to, because a monopoly on payment processing makes a lot of money.
If 4%/15%/30%/whatever is what Google needs to charge to run their payment processing business, that shows that they're extremely inefficient/incompetent, and the market would be more efficient if other companies were able to offer that service instead. Last I checked, Stripe charges 2.9%, which is even less than what Google charges Spotify in this unicorn deal.
EDIT: the same goes for "store infrastructure" or whatever. If Google offered publishers the option to run their own infrastructure for distributing apps/updates/whatever, everyone would take that deal rather than pay Google's nonsense fees.