The magnetic moment of the nuclei being read by the MRI instrument does have a stronger coherence when the magnet is stronger - the energy difference between the excited and ground state of the atom's nuclear spin is larger.
This does lengthen the relaxation time as you say, but the really important effect is that the resolution of the MRI instrument becomes greater. When differentiating between nuclear spins of protons in cancer tissue vs healthy tissues is very small - but fortunately it is there. Most of the medical applications need as much resolution as they can get because there is a massive amount of protons in water or phosophorus the machine has to differentiate over - most if the 'signal' is miniscule compared to native state biological atoms the MRI is also seeing.
@DrSAR I'm sort of confused by your comment. Not sure what you are saying.
1) MRI is one of the more common ways to identify cancer. Maybe if you could be more explicit in why you do not believe NMR (I use this term interchangably with MRI) can differentiate physiological state of tissues? This has been pretty well discussed since the 1980s.
this quote passed wikipedia's editorial process okay:
MRI provides good contrast between the different soft tissues of the body, which makes it especially useful in imaging the brain, muscles, the heart, and cancers compared with other medical imaging techniques such as computed tomography (CT) or X-rays. Unlike CT scans or traditional X-rays, MRI does not use ionizing radiation.
2) MRI does not need protons, any nucleus with a spin =1/2 is a decent target for an experiment. again P31 is one of the more popular targets because it has a high natural abundance, but O17 and C13 are also accessible through most machines (and their high intensity magnetic fields) today. More exotic spins (3/2 5/2 etc) can also be used, but the sensitivity and analysis of the returned signal are more complicated. So many different kinds of atoms can and are used in MRI experiments.