Math can be an end unto itself. This can come as a bit of a surprise in our prevailing culture, which needs to justify the usefulness of everything. Also, it's possible for someone to study math as a liberal art, and develop the ability to do useful things with it on their own. My observation is that the people who grudgingly learned math as a means to an end, tend to forget most of it soon after graduating. This explains the widespread but paradoxical aversion to math among engineers.

Which is exactly what Hardy said himself.

Utility is not the ultimate goal of math, understanding is.

Not becoming a patient-treating doctor. Research doctors still matter a great deal in the field of medicine.

I doubt you have pure research doctors. Medicine is a field that is so dependent on treatment outcomes. There will be doctors more focused on research. However, I doubt they will stop seeing patients.

I know for a fact that pediatric oncology and hematology is entirely driven out of a research hospital or university. But doctors there publish but also treat.

There are lots of medical researchers who don’t treat patients.

I think that would also be rewarding. I have no desire to become a doctor but I'm interested in how medicine works.

On the contrary, it's more like studying medicine and then remaining in medical research in a purely academic (non-clinical) setting.

John Keats, Osamu Tezuka, Somerset Maugham, Hector Berlioz.

Studied medicine but did not practice (Keats did for a little while). Just for interest.

Also, Che Guevara.

so like studying (human) biology and becoming a biologist