In most parts of europe health care is not payed with taxes.
On top of that the "systems don't work" (actually they work) because there is no real market. Different "institutions" have different interests.
E.g. in Germany hospitals like to have people to long in the hospital, as the hospital is payed per day.
That has nothing to do with immigrants, low taxes or non working refugees.
In Denmark health care is payed by taxes, it is basically a branch of the state. You get kicked out of the hospital as soon as you seem fit.
Bottom line the system is the same: people are required to pay as an "insurance" money into a pool. That pool is used to treat those who need.
Nevertheless the way how it can be done and how much you bottom line pay is quite different.