Yes, and this can be done by software firms. You don't need to roll your own solution, but taking the payment networks' fees out of the equation drastically reduces costs.
> I'm not aware of any payment provider that gives me all European networks, but not credit cards.
That's true, since most sellers can access credit cards. But for EU-only sellers of "sensitive products", of course there is a market. (It's an open question if people on the MATCH list can use those payment providers as long as they don't accept cards.)
> Together with 95% of your customer base, realistically.
Most people where I live are comfortable using mobile payments and use it on a regular basis. I see mainstream retailers offering to take it (alongside cards) very often.