IMO testing the caps is a good idea : look for moisture around the caps on the pcb and/or even better measure them with a multimeter.
I would (since you have multiple boards) consider buying a MATT card to test the boards. It's a SPI flash (quick and dirty BootROM) you can plug in to the Logic Board (ignoring the SPI flash on the board).
@tsialex is the expert on this topic. If the boards work with a MATT card, you could ask him to restore the BootROM and flash it to a new SPI flash you could solder in.