His treatment was controversial. The doctors removed the bullet and probed the wound with unwashed hands. It is not certain he would have lived much longer if left less attended, but the doctors, on balance, probably did more hard (unintentionally) than good. Some speculate he might have recovered.
It’s not exactly controversial. The bullet entered the rear of his skull, transited the brain and ended up behind his eye. I’ve read a number of books about the assassination and none seriously contended there was any chance of his survival.
In particular, there was no way unwashed hands contributed to his death, since there was not sufficient time for this to have caused infection.
Here’s an article by a trauma physician who believes he would have died even with modern trauma center or battlefield emergency care.
http://showcase.netins.net/web/creative/lincoln/education/medical.htm
I wonder if you’re confusing his case with that of Garfield, who would almost certainly have survived with proper treatment.
Also, the bullet was not removed till the autopsy.