Mike Paulsen wrote:

You could install a 36" 1x4 (oak or whatever matches the cabinets) and
then attach the towel rack to that.

It would cover the damaged drywall and would be plenty strong. (you'd be
able to hit three studs)

I had a similar problem with a light fixture above a bath mirror. The box was
improperly installed and there was a vent pipe in the wall that made reinforcing
that area extremely difficult. Ran a 1/2x6 the width of the mirror just above
the mirror such that it almost looked like the mirror was framed in, anchored to
2 or three studs, then put a retrofit box in the 1/2x6. Worked out quite well.