It's been a minute since I have removed a pilot bushing/bearing. I too learned on grease and a socket snapped on backwards to an extension for a pusher. It has to fit real snug and I kind of remember wrapping electrical tape around the socket to make it fit tighter. I can see where the bread will work. Particularly white bread. We used to smush it into a gooey ball and put it on a fishhook to catch fish. One time I didn't have any grease and resorted to packing the hole with leftover RTV. It setup for a few minutes then it popped out just like using grease.
The last bronze bushing that was removed at our shop, brother drilled a pair of 1/4" holes in it to weaken it up a bit and it popped right out with the slide hammer tool. Yes, bronze is way softer to drill into than what the OP was attempting to drill into, the bearing with the steel outer race of the old roller bearing.