A system can be private and be funded in any of those ways. The incentive to provide privacy is that it allows them to out-compete other payment systems without privacy, and then they get more deposits/transactions/unclaimed funds.

The primary reason this doesn't exist isn't that companies wouldn't provide it, it's that there are various laws that effectively prohibit transaction processing systems that provide meaningful cryptographic privacy guarantees.

This is where a lot of people had hope for blockchain -- most of those laws apply to the payment processor and it doesn't inherently require any payment processor entity for them to apply to. But with that comes a lot of other disadvantages (limited transaction volume, wasteful energy consumption, lack of a controlling party to provide up-front resources to address early user experience issues), not to mention the speculation-inducing deflationary nonsense, which would have to be overcome before it could expect to become popular with ordinary people.

Enough of those might eventually be addressed enough to make such systems enter common usage, but we would obviously be better off to do something about the laws prohibiting cryptographically private payment processing systems using ordinary payment processors instead of forcing people onto these ugly workarounds for something everybody should have a right to anyway. It's still not any of the government's business what books I read.