I have not seen many failed premeds who are good at math. In fact, I would bet for those students who are good at math, they can excel at premed even if they start late (just like @WayOutWestMom 's daughters). But not the other way around. (those who are good at math might quit premed voluntarily but unlikely would fail if they choose to continue) Not everyone is interested in medicine for sure. If someone has been volunteering in hospitals for more than 200 hours and still like it, I am confident the person quite possibly will like it.
There are many different working environments for people in medicine. If you bring your kid to experience “The Alexandria Center for Life Science” (http://www.alexandrianyc.com/ersp.html), I am confident your kid would be at least interested in the life science for a while. 
But you bring your kid to a morgue to volunteer, I am not so sure your kid would like it. (this “MIT girl” did just that. https://www.youtube.com/watch?v=7PthOO6XkIc She went to MSU and quit premed) There are many faces for every career. One advantage of being in medicine is that there are so many specialties. If you don’t like to see patient die on you, choose one specialty so it is unlikely to happen. (but if you faint after seeing blood, you are not for medicine) For whatever reason, there are more doctors who want their kids to be in the same career path than those who work for other jobs. I have not figured out why completely but have some idea.