Why would a Christian go to a Muslim doctor?
Why wouldn't they? In terms of, say, a root canal or getting stitches, a doctor's religion would be wholly immaterial (and would never even come up in conversation- I have no idea what religion, if any, my doctor follows)
"Why would a Christian go to a Muslim doctor?"
In an emergency, we don't get to choose who treats us. Whoever is working that shift in the ER is the one who treats us.