Logan: Go fuck yourself, pretty boy.
Yukio: You're a soldier, and you seek what all soldiers do.
Logan: and what's that?
Yukio: an honorable death, an end to your pain.
Logan: I can do this all day, you twisted mutant bitch!
Logan: A lot of people have tried to kill me... And I'm still here.
Yukio: You look nice.
Logan: I feel violated.
Logan: Is that all the men you brought?
Yashida: Eternity can be a curse. The losses you have had to suffer... A man can run out of things to care for, lose his purpose.
Yashida: For so many years, I have wanted to thank you for saving my life. But I didn't send for you only to thank you, Logan. I wanted to repay you: a gift to equal the life you gave me.
Logan: What they did to me, what I am, can't be undone.
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.