How to repair a hole in an attached garage

I'm in a temp range about 5C more in each direction. Attached garages tend to be about half as cold as outside so a thermal break is less important here. It wouldn't be a huge loss if you just screwed the gypsum board directly to the underside of the floor joists. However, it's easy to get a thermal break without much more effort, and it's almost as easy as applying directly to floor joists.

After applying the main insulation, sheath the bottom of the joists with 1/2" polyiso (polyisocyanurate) insulation board -- your thermal break. Then using longer screws, apply the gypsum directly to the joists through the polyiso.

Highest possible ceiling, thermal break if you want one, level application, no faffing about with hangers. Wins all around.

I would not waste the insulation on an outside wall unless your garage door was getting insulation. Drywalling has benefits though... First you need to use fire resistant (5/8) in most places in the US for garages, a fully drywalled and painted garage will be deemed nicer by those wanting to give you money for your house, white flat walls reflect light the best, noise reduction... all I could think of.

Small tip if you do this... Run some crosses at like 5 and 7 feet along the wall. That way you have places to nail things without measuring for a stud.