Car won't turn over on good battery and starter

The car is a 2007 Mazda6 3L which I parked overnight and won’t turn over in the morning anymore. I did some preliminary diagnostic and ruled out some possibilities:

• The battery is full, 12.4 V.
• The starter is good, so is the connection from the load side of the relay to the starter. I was able to jump the starter powering the relay.
• The ignition switch is good, I hear a click even with the relay removed.

When the starter is jump started, the engine shakes but it still doesn’t start, so I’m think maybe the fuel injection system is locked by safety feature. I’m thinking the neutral safety switch, but had no luck starting in other gears as well. Is there anything else I should check for?

Edit: There was a u0100 code that popped up, has to do with lost communication with the ECU. Not sure if related.