If the extractor design pushes against the gear, the sprocket assy could be pushed deeper along the tube.Or .... Not good.
I agree with you. Just use the same tool that has worked thousands of times for this specific and unique task. Guinea pig Threads are always interesting and cn be valuable when new parts/tools/techniques are shared.
Good luck with your bearing choice.
Long ago there was a disaster Thread on this subject where an infamous Poster used a self-selected bearing against the advice of the Bearing distributor. If you search "Boca Bearing" it may come up.
Here it is. A classic , best-of-intentions goes badly wrong Rennlist gem.
All I am doing is trying to help the P.O. make a fully informed choice about the many gotchas in this subject.They have all been covered in scattered Posts here but maybe it is futile to help.
https://rennlist.com/forums/996-forum/929113-i-bought-kk-s-car-long-story-5.html
Finding a good bearing substitute is easy catalog work.
Finding and installing one that works well is another story.