Assuming open flame heaters are allowed in a garage in your area, carefully check all clearance requirements for the stove and flue. I would encourage you to adapt to triple wall insulated flue pipe at the ceiling of the first floor and stay with insulated pipe the rest of the way. Do not attempt to change back to single wall pipe.
You may want to install a couple of floor registers to help take some of the warm air from the first floor to the second floor. Remember, heat rises, and depending on how long you run the stove, the heat will eventually fill the second floor. Some simple electric fans can also be placed strategically to move the warm air around and even out the temp in the building.
Nit picking on terminology the roof (or the closet drywall) will be damaged. That's what cutting a hole is. What you need to concern yourself is will it be repaired properly.
Basically a roof consists of the roof framing, the sheathing and the weatherproof covering.
The framing should not be cut in the roof. It just complicates things too much and isn't necessary. If it accidentally does get cut it can be repaired.
The sheathing depends on when your home was built. Decades ago you may have seen solid wood boards such as 1x6's, later plywood and more recently either plywood or OSB. Plywood and OSB are easier to replace and more people will have experience with it. If it's an older home with solid boards as the decking you may want to avoid messing with it.
The covering, things like asphalt shings are fairly easy to replace. If you have something like a clay tile roof then this would be more difficult but I doubt the contractor would have suggested cutting through the roof if that were the case. I think it would be best to use new shingles when repairing the hole but I'm not a roofer. With new shingles you'll have a patch that looks different than the rest of the roof.
Cutting through walls is similar in that there is framing, drywall then drywall compound over seams and finally paint. Much easier to repair and if something goes wrong with the repair you don't have to worry about a leak.
Going through the interior closet is the safer choice for you, going through the roof probably makes attic access easier for the contractor and could be less messy for you since all the cutting and running of insulation hoses happens outside.
Since your walls are plaster it could make the repair more complicated but the repair can also be made with drywall, shims and maybe a little extra mud to even things out. It may not come out as could than properly repairing the plaster but in a closet I wouldn't mind too much myself. The other option is to cut out a proper sized attic access and install a hatch. To be code compliant the opening is larger than the 16" OC the ceiling framing probably is (I believe minimum is 24"x30" but double check). That would involve adding some framing around the hole to accommodate it. Not that big a deal but might be something to consider if you'd like to have access to that attic area in the future. The larger hole should also make it easier for the contractor to bring up insulation.
I personally would feel more comfortable with cutting a hole in the drywall than the roof. If you like the contractor/equipment/insulation from the guy that wants to go through the roof you can probably tell him you'd rather he go in through an interior wall and not make any holes in your roof. He'd probably agree to it.
Best Answer
No. That would be like sealing the bottom side of a sponge--moisture will travel through it to emerge in other areas.
You can probably accomplish the repair even in winter. A water hose will quickly melt snow and ice from the areas you need to access, or use scaffolding and other hardware to work over it.