What is the proper method for repairing insulation nicks in wiring?
Electrical tape can insulate a splice all on its own, so this should be fine

UL-listed thermoplastic electrical tapes are intended to be the sole insulating means on a splice or joint (the White Book says as much), and are tested for their capacity as an insulating material, so using them to repair damaged wire insulation within a junction box should be fine in practice, even though neither the NEC nor the UL White Book speak directly to this use of the product.