Proper bottom bracket service means installing on a shell that has been correctly chased and faced, using a bottom bracket tool, not a wrench.
https://www.hozan.co.jp/cycle_e/catalog/BB/C-358.html

This is also the proper tool for removal.

A bottom bracket that has been properly installed as above should not have its fixed cup removed for regular servicing (repacking). BITD people didn't have 5 bikes. Racers and enthusiast cyclists rode 300+/week. BB bearings got repacked about twice a year on average, more if caught in the rain. Fixed cups were not removed until the bottom bracket was worn out and replaced. It would have been silly.

An old bike of unknown history that's been sitting for decades generally should have both cups removed for servicing.