If you want to do that, then you have to start charging the middle and upper class several thousand dollars a month. That would quickly destabilize the (already fragile) economy.
Btw, I'm in the lower class.
I don't know the details of how taxes are staggered in the US but, no, you don't. Nobody in the UK is paying several thousand dollars for their National Insurance (not unless they're earning hundreds of thousands a year, anyway...)
Because everybody is paying into it, the cost per head is lower to begin with. And many people who are paying in will have alternative healthcare arrangements anyway, and therefore won't make use of the free service. So you can further reduce the amount that needs to be collected. Even little poor old Jim is still paying in his 35cents a week or whatever.... and of course he may never even get sick (after all of our worrying..!) and therefore not need the service - and all those Jims out there paying their 35 cents.... they all add up.