Visible crew/equipment: In the funeral scene where Logan is attacked, he gets shot and knocked down. The camera pans towards Logan and we get a close up of his exposed claws. If you look in his fist you can see Hugh Jackman is holding on to a round gold colored object in plain view. This is the handle Hugh Jackman holds on to for his fake movie prop claws.
Visible crew/equipment: When the bear appears as Logan goes to buy batteries, a cameraman with a brown jacket can be spotted at the lower right hand corner, before the shot changes.
Answer: The healing factor was only reduced. Like you said, had it been removed he would have bled to death and every time he used his claws he would be bleeding from his hands. It is not revealed if it is at full strength at the end of the film, but it is likely it will return to full strength over time.