I think your Wallet.pay(amount) example argues for a different conclusion than you arrive at.
I have a Wallet with $500 in it. I call Wallet.pay(100). In your approach, it returns a new Wallet with $400 in it. But I also still have the old, unmutated Wallet with $500 in it, which could be referred to by mistake (or by malice). That's probably not the best argument for immutable objects...