In situations where cauterization is recommended in nostril to prevent recurring nosebleeds, can the body eventually mend this problem on its own?

Emollient. Agree with the answer given by Dr. Pool. You may help yourself by applying a very small amount of an emollient, like Vaseline, to the opening of the nose, on the inside. This will prevent excessive drying of the area that generally bleeds. Be sure to not over do the emollient.
Yes but. Most nosebleeds can be controlled with direct pressure but some types will tend to re-bleed easily. Consult your ENT doctor for the best strategy to prevent recurrent bleeds. But all of them start with not picking your nose.