1) Jesus did NOT use the state to provide healthcare, nor did the NT church...rather they taught individuals to provide for the poor.
Would you care to cite a verse for us where Jesus said the state was not to provide healthcare or otherwise provide for the poor?
I'd like to point out that in Leviticus 14:21-22, the offering that was to be made when a person was cleansed of tsara’at was lessened for the poor, (as were other offerings in other circumstances as well). It would appear to me that God is in favor of adjusting charges for health care downward for those unable to pay the the full price.