Your card provider needs to keep you happy far more than your card provider needs to keep Uber happy.
With Uber, there simply isn't an alternative. If they don't accept Visa then i can't use a credit card (or most debit cards, since those go through the visa network too) and there isn't a reasonable way for me to pay at all.
Uber takes Paypal, Venmo, Apple Pay, and Google Pay too. Any would let you use a bank account. You can also deposit cash into PayPal and I think Venmo as well.
They (in collusion with MasterCard) could jack up prices and then there's be an antitrust case against them.
Here's the thing, though - as long as they don't abuse their duopoly position, those anti-trust cases are weaker. And a policy of preferring to side with the customer in these disputes is not exactly viewed as abuse of the monopoly position.