Different Mohs surgeons treat their 'OR's differently. Not many will 'scrub in' the way you have experienced in surgery as an M3.
There are procedures such as liquid nitrogen along with random snips, clips, and other things.
ACGME requires us to log our procedures during residency. So you cannot get through residency without doing them. We need 50 surgeries with complex repairs, 15 or so laser experiences, 3 nail procedures, 15 or so botox, etc. Some of them only need to be observational. The 50 surgeries need to be done by the resident though.
I'd say that I'm doing a biopsy on 1 out of 10 or so patients. So it happens pretty often. Add in other procedures such as LN, and that ratio goes up.
You can probably end up in a derm niche where you don't do biopsies much. However, it's unavoidable in residency. If you really dislike procedures/surgeries, you may consider something else. Maybe Rhuem or Allergy. They have procedures, but not so much of cutting type stuff (like biopsies). However, you will have to do lines and such during most any residency related to medicine.