While I'm not in a position to assess the risks, and unintended side-effects have been documented, e.g. unexpected elevation of mood in subjects receiving cranial MRI, I will note that the "nuclear" in NMR, of which MRI is an off-shoot/specialization, refers to the structure being imaged rather than to a nuclear reaction with the latter's attendant risks.

I remember when this technology was "up and coming" and "NMR" was the term/acronym. I wondered a bit when "MRI" started cropping up; perhaps you are right that "nuclear" was a bit scary for the general/patient population.

https://en.wikipedia.org/wiki/Nuclear_magnetic_resonance

https://en.wikipedia.org/wiki/Magnetic_resonance_imaging