Reading through all the comments and have an additional thought while you're waiting for the SSD to arrive. You mention you have cleaned up the board and reseated all the connections/made sure they are sound.
Have you inspected the trackpad? Since it's under the battery it often gets overlooked.
Check on the trackpad side of the cable for any signs of liquid damage, or even if the cable is loose or crooked. It could be that it's slightly askew in the connector. See what happens if you try and power it on without the trackpad cable connected.
A bad trackpad cable, or minor liquid damage or debris in the connector, can cause some weird behavior. The trackpad carries electrical data tied to both power on and wake from sleep functions. And it's incredibly vulnerable to damage since that connector is close to the edge of the trackpad. And there is nothing to prevent debris, or liquid from getting in.