Thanks.
Wait, the claw of the hammer is under a screw. There is no screw there on ours, and there are no threads on the inside of the handle part of the cartridge. Is that a sheet metal screw, or something similar that will cut its own grooves?
Anyway, mechanically that is pretty close to what I ended up with. Took the 11" pry bar from this set:
https://www.harborfreight.com/2-piece-flat-pry-bar-set-67477.html
placed the pointy end of the 90 degree claw on the cartridge body below the flange, which was really beaten up by this point and had been pried up about a millimeter so the body was exposed. Tapped it in gently with a rubber mallet so that it "bit". Put a 2x4 underneath that and pulled away from the cartridge and down. It took a couple of attempts, but eventually the cartridge popped up about a centimeter and was free. Interestingly (in terms of the next replacement) the new one dropped in and stopped about 1 cm up. More or less the point where it became "free" when pried up. It had to be tapped down gently until the nut could grab some threads and pull it in the rest of the way.
Things that didn't work for me:
1. Turn the water on. The cartridge didn't budge and it didn't even leak. Our house pressure is 80 PSI.
2. Clamp a vice grip onto the handle stem part of the cartridge, put a screw driver in above the handle, inside the clamp, pry up. This might have worked but I didn't have a really big flat blade screwdriver that would provide enough leverage. My biggest flat blade is only around 8" long.
3. Rotate the cartridge. Grabbed the flange with a big pair of channel locks and rocked it back and forth. Eventually it sheared off the tab and the cartridge could rotate (with some force) a full 360. But that didn't loosen whatever was holding it in. The basis for this was experience removing car hoses, where rotating the hose on the pipe a few degrees back and forth is often enough to break it free. Not so here.
Got to laugh at the illustrated instructions that came with the cartridge. It shows a small set of pliers, probably 8" long, used to pull the cartridge out by the stem. It doesn't show the gorilla holding those pliers!