FWIW, Stripe.com supports ACH (Automated Clearing House electronic funds transfer in the US) as well as payment cards. So a SaaS can take that kind of payment too, without lots of hassle. I haven't investigated Stripe's worthy competitors' ACH offerings, but they surely have them too.
And, big company purchasing departments can definitely get their hands on suitable payment cards for this purpose. They can guard them closely, and don't have to hand them out to random people.
This problem can be solved. (Well, it might take a decade or two...)