I have found that getting old bearing cups out works best by using an old seatpost or a piece of pipe instead of a screwdriver.....Yours are the worst I have ever seen......Nice find on that frame!!!....lots of possibilities there!!!
Oh man, rusted seized seatpost especially that one smushed way down is the worst. Looks like lots of post surface area inside the tube. Perhaps make vertical cut in post running from between seattube slot up to top of post, like with dremel cutting wheel or similar. Squeeze post betwn vice or similar. Resulting flex should free up some area. And can drip some solution into the gap too.
Or if you have long cutting blade, carefully cut vertical line inside post (careful to not cut seattube) and then squeeze to break free as much post surface contact area away. Then solution soak and twist as usual.
I've done this with seized interfaces with success. But not smushed down like yours.