There just isn't one tool made for this job.
Use the largest diameter rod you have that will fit inside headtube or cup, insert just to meet the edge of the rivet tail, then lift and push to other edge of rivet, pry against opposite side of headtube or cup to push the tail of the rivet out and flush with the inside of the headtube. I think even a seatpost will work for this step, 1-1/8" solid rod probably works best. Next, I use a cotter pin extractor, find the center of the twist rivet with the pointy L shaped end of the cotter pin extractor and pry in the same fashion as the rod. The pointy tip is usually enough to push the rivet all the way out, so be prepared to catch the shooting rivet! Finding the center of the rivet tail and staying on it is the hardest part. Not all cotter pin extractors are created equal either, so you may need to add a flat shim behind the bent part of the tool for better leverage against the inside of the headtube. Some rivets are tough too and you may have to tap the top of the cotter pin extractor with a hammer. Holding on to the shim, extractor, and hammer while catching the rivet can be quite the juggling act...so, a piece of masking tape over the front of the rivet comes in handy to keep it from flying.
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