Shaving on Chol Hamoed

May one who shaves on a regular basis, shave on chol hamoed as well?

Good question, but no.

While the reasoning you stated would not allow a carte Blanche approach to shaving on chol Hamoed, nevertheless in certain circumstances it would be permissible. The Rav has allowed it when, for example ones child has a bris (brit) on chol Hamoed or other such “tzorech”. The determination of “tzorech-need” should be determined by consulting your Rav.