I think it's less a philosophical question -- is mail like roads or police? -- vs a matter of efficiency. It's obviously possible to NOT lose money delivering much more difficult things than paper. FedEx and UPS do it all the time. But they're private businesses who compete with each other (and Amazon and whoever else), so they have to figure it out or they won't be around.
The USPS, meanwhile, doesn't have to compete with anyone (in fact, it's illegal for anyone else to deliver mail) and has no incentive or need to be efficient because they can draw on the US Treasury whenever they need to. It's not that they're bad people, it's just the nature of the incentives and system involved (and also it's run by congress). These same dynamic would be there in banking though, which is why this seems like such a bad idea.