We already compel private behavior all the time when the private sector starts acting public - i.e. by running infrastructure and being vital to the function of the public.

The real solution IMO is even more unpopular: nationalize them. If it's a public service it should be handled by the public sector, such that the entirety of the constitution applies. We might even consider funding it not with payment fees, but tax dollars. Every American has a desire to have reliable instant transactions. So they should all pay.

Effectively, they already are - the 2-3% tax on card processors is a tax. If we nationalize it, we can even lower it, since we'd not longer be burdened by the pursuit of profit.