it's called "concierge" care. It's fine if all you ever need is a primary care doc you can talk to on the phone and who will even make house calls (these docs do typically take some low-income patients on a sliding scale or pro bono), but it doesn't do anything for people who need specialists, or hospital care, etc.
(Nor would it help in an emergency or unexpected situation requiring expensive care which, as Val said, is the point of insurance)
I haven't seen it break out of primary care yet. I could see certain specialists adopting it, but I kind of doubt that it would work on a hospital level.
If we want to solve healthcare, we have to fix the hospitals and multispecialty clinics, as well as the device makers, pharma companies, etc.
Also break the tie to employment. The user should make the purchasing choice, not a third party.