This is exactly how it works in the UK - universal coverage of everyone via the NHS, and additional elective procedures covered either via private insurance or direct payment at private facilities.

I actually think that government healthcare might be a good opportunity to reduce regulation of private healthcare: if the government is paying for it, they dictate the terms, but for a private transaction, much less oversight would be needed. Of course having this happen would require compromise which appears to be completely absent from our current legislative process...

That’s how it works in most countries. People with more money can buy better treatment.

Are electives not covered by your insurance? Do you pay for them entirely out of pocket?