What you describe is pretty close to what we have in Australia. Free public hospitals for all, but people are permitted to pay extra for private medical services if they want. Mostly that just means slightly nicer decor in the doctor’s office or hospital.
However, the effect is as described upthread; the wealthy constantly attempt to defund and downgrade the public hospitals, as they don’t use them. They’ve been trying to destroy the public health system for as long as it’s existed.