That wouldn't make sense, since in one of the endings Geralt is retired. Besides all the witchers at Kaer Morhen were against the idea of making new ones. And even if they wanted to, they don't have means to accomplish it.
Geralt playing the mentor role would make perfect sense. He's retired, that means, he's not traveling the World getting caught up in everything. That doesn't mean he's just gonna sit around KM or some cabin in the woods some day. I can totally see him getting a bit bored and becoming a mentor/trainer to a new character.
And while the witchers of KM may be against the idea of training witchers, they and the rest of the world can come to realize that it's needed. As of W3 there are no known schools currently training witchers, and with witchers retiring and/or dying, that means the monster population is gonna rise and soon people are gonna come to realize that witchers or some new order that can effectively kill these monsters are greatly needed.
A big part of the plot can be the conflict of the raising and training witchers. Yes, the process is brutal, between the training and the mutations, not many survive and then not a lot of those that get through all that even make it very long on their adventures of traveling the lands and slaying these monsters.
A good part of the story and quests can be looking for ways to alter the mutations such as the trial of grasses to make it a lot of safer with better chances of survival and even try and get these new formulas to other schools so they can train better witchers too, and also in cases like that Cat School, they can get rid of a lot of nasty side effects such as turning insane and sometimes killing everything they see.
I can totally see the player character looking for clues, finding old knowledge and consulting mages and looking for druids like Ermion or alchemists that would be willing to help.