Nope. In your scenario everyone is covered yes? We just increased demand. Are we going to magic more doctors/hospitals/drugs out of nowhere? No. So supply stays the same. You can not increase demand and quality of care while leaving supply alone and expect costs to decrease. That is not real world.
It has already been explained to you how the “futile” care the US engages in makes our numbers look worse. We try and save our Grandmas and our Charlie Gards. Countries with rationed care don’t.
We’ve already established that nearly 2/3 of healthcare spending is already done by government, making that 100% will somehow improve care and costs eh?
I’d like to point out something else. Scale. Every country held up as an ideal by the socialized medicine crowd is far smaller the US. For example the total budget for the NHS in the UK is roughly $163 billion. Medical fraud in the US costs $272 billion. Our fraud costs more than their entire system.