There are 5 different power states of a MacBook Pro, and different circuits on the motherboard are responsible for these different circuits. The battery definitely seems to be charging (since the light changes colour), and the SMC and associated circuits seem to be functional since the SMC reset is working. So, the G3H and S5 circuit which are the "shut down" states work fine. That means that there is most likely an issue with an S3 (sleep) of S0 (full on) circuit of the motherboard. Have you spilled liquid on your computer? If so, remove the logic board and clean with 99% isopropanol.
The power button is not the issue here as this logic board turns on when the charger is connected automatically when the board is working properly, so no need to use the power-on or jump start pads. A bad battery will never be the the cause of a board not booting. I don't know what Shawn is talking about - there are definitely no "pressure sensors" on the board that requires the bottom case to be seated for the board to turn on. The board should turn on even if it's out of the laptop with nothing but the charger connected.