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Depending on the bike, you can end up with both the old and new bearing wedged in doing that, and the only way to get them out involves damaging both, that's why I use a socket!
As another option, have you tried inserting a finger into the bearings with the spindle removed and seeing how much play there is in the inner spacer, there was always a little in that it was never a real tight fit. If yours is quite loose it would point to the bearings not being fully located.