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If the top bolts pulled through the transom, it is likely not just the area around them that is rotten. What you are suggesting sounds like a good quick fix, but the next thing to pull through the transom will likely be the Git-Rot along with the bolts, and it will make a bigger hole. I would suggest doing a core sample of your transom to find out how bad it really is before making any decisions on a fix.
Yeah, a prybar, and a hammer can work wonders, but still a lot of work, any way you slice it.
Not a lot of marine plywood suppliers carry >3/4" ply. Sandwiching multiple panels together is easy enough. For any future projects I would recommend AquaPlas III Transom board. It is a PVC foam that is a good match for plywood as far as density goes. It has excellent screw retention, and we can even CNC Cut it to the shape of your transom in multiple layers, to glue together (product is 1/2" thick).