This is the most strawman of strawmen I might ever have heard.

By your logic, the word "free" should be banished from the English language, because literally nothing could ever be free.

Except, for people who have common sense, "free" means you don't have to pay for the thing directly.

In this case, your taxes get aggregated with everyone else's and then some gets split up into health services. But since there's no direct connection between the two -- you don't get more healthcare if you pay more taxes individually, and you don't get to pay less taxes if you don't go to the doctor -- it's conceptualized as paying taxes on one end, and getting free health care on the other. This is just common sense. Everyone understands how this works. It's the same way we have free schools. Or you think schools aren't actually free either...?