I haven't read the full text of the bill, but presumably not. The intent will be to disallow private coverage for any service that the government would provide, specifically because they would want to make sure that people with money couldn't avoid the public system.

That's how it ends up working in many places with government care

I don't know of any healthcare system where what you're describing is the case outside of outliers like North Korea. Even in the UK with NHS, there's a whole system of private insurance and private hospitals too.

In Norway and UK there is public healthcare, but you can pay to skip the lines if you want to. I don't know why we would go full socialist in the US when living in a capitalist society. Should we only get cars allocated to us or something as well?

I am not hugely in favor or capitalism, but I can't see anything better or a way to replace it even though I am a social democrat. Let people with money spend them as long as we provide a decent floor for everyone.