MacBook Pro won't turn on with power button on keyboard

I inherited an early 2013 Macbook Pro 15" that wasn't starting up. MagSafe lights up, but no power. Tried disconnecting and re-connecting the battery. No change. Tried disconnecting the battery holding power button down to fully drain the power. Once I reconnect the battery, as soon as the MagSafe adapter is connected, the laptop starts up and functions perfectly, aside from the power button functioning. If I shut down the laptop, I can't power it back on with the power button. I need to disconnect the battery, hold power, reconnect the battery and attach the MagSafe, then it powers back up again on it's own.

The power button must be physically working as I have to press it to fully drain the board, before plugging it back in, and I have already wiped and re-installed the OS with all updates, and I've reset the NVRAM. I tried doing the SMC reset, but the power button doesn't appear to function during that.

Obviously this sounds like a board issue, but is there anyway to manually flash the firmware on one of these, or anything else I can try to do to resolve this? Laptop seems to work fine otherwise.

EDIT: Just realized it's actually a Late 2013 model not the early 2013 one I originally posted

Thanks