There are ways to punish unvaxxes without blocking them from healthcare which is deadly and a ethical misstep. And smarter ways to enforce people into vaccination without directly sending them to shots
Companies should be forced to fire non vaccinated workers or suffer legal responsibilities, and public spaces to require either covid tests or vaccines from customers and visitors. This would make public life near impossible. In Brazil unvaccinated people is already being blocked to attend public spaces (in few cities where vaccinated is advanced)
And of course, unvacinneted families should be responsible for paying all expenses for treatment. That's would never be a thing in USA as your system is already private, but I'm sure would be effective for countries with public healthcare