It's "bigger" in the sense it spends more money per capita. Something very American exceptionalist about the OP suggesting that this is somehow more relevant than it covering fewer people and treatments.
I am not sure what your point is about covering fewer people either. The point of public healthcare systems is that there are redistributive, correct? The reason the US public healthcare system does not cover everyone is because there are people who can pay for their own healthcare...which is the same in Europe. I live in Europe, in a system with "free healthcare", I pay $100/month for private healthcare because queues for most things are multiple years long AND I pay $1-1.5k/month for other people to use the public healthcare system I can't use.
The irony is in the US rather than paying a low amount on private healthcare so that you could potentially beat the queue on relatively low cost state healthcare, you would be paying a much higher rate for private healthcare with more copays and coverage exemptions and also contributing a higher portion of your taxes to a publicly funded insurance program you wouldn't be able to access at all when you lost your job or your insurer denied treatment.