Are you going to get close enough to apply first aid to your assailant? Are you going to give them a second chance to do you harm?
It depends on the situation at hand. Even cops do it both ways. I've seen cops give first-responder type treatment to folks after they're handcuffed and no longer a threat, and I've seen cops wait until backup arrives if the guy is still resistant. Granted, we've got handcuffs and the application of the cuffs generally means that active resistance is over, but there are times when we don't have handcuffs and we're still obligated to take the person into custody and provide first aid.
That said,I've arrested a person when all I had was a roll of duct tape. That makes a very efficient handcuff. The rule of thumb for us is, if the person is still resistant, we cannot administer first aid until they're in custody. Once they're in custody, we have a responsibility to administer first aid.
Depending on your state laws, there might be an obligation to assist once the threat has ended. There might not be. There might also be certain legal presumptions regarding the obligation of the victim to, at a minimum, notify first-responders to administer aid.
But, I would agree that all persons over the age of 10 or 12 years need basic first-aid training. Stop-the-bleeding, Heimlich, and CPR as a basic minimum.