Almost the exact same but I now use a brass rodI also hammer them out, but use something heavier than a screwdriver. Also more of a blunt striking surface. I have a section of 5/8ths steel rod leftover from making seat posts that works great to knock out cups on the headtube also. As long as you don't damage the race it should be fine.
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Well you can make a dedicated tool out of a piece of pipe. You need it to be a little less in diameter than the BB. Then cut one end in a cross pattern, maybe four inches, this will make the ends spread a little so they will rest against the BB cup and when you hit with the mallet the force will be distributed evenly.
Glens bicycle repair manual, the old bike repair bible, from the 60s - 70s used a punch and hammer. I don’t think all these fancy tools were around then. Good enough for Glen, good enough for me.