First of all get that battery off of the MPU. It's a time bomb. The cheapest replacement would be to use a 4 place AA battery holder and mount it in the backbox. Install a diode in one of the battery compartments.
Another option would be to replace U8 and U13 with NVRAM's.
The rectifier board is repairable but NVRAM.WEEBLY makes a nice kit that easy to assemble and don't cost much more than the parts alone.
Your boot up problem may be with the connectors. I'd suggest you replace all the connectors and headers on all the boards.