The green light on the motherboard doesn't necessarily mean that the full motherboard is working, as you say it is an indicator that it is receiving power from the psu. Often these power led's aren't connected to the rest of the board so doesn't really tell you either way.
If the CMOS battery has a burn mark on it I would recommend that you replace that, damaged lithium batteries are a serious fire risk (think Samsung Galaxy note). Once you have replaced it you may find that it solves the issue. There are forum posts around that say it can solve the issue you seem to be experiencing.