If you are able to start it by jumping the starter, then the immobilizer isn't the issue. Have you tried retrieving diagnostic trouble codes? Also, did you try starting it in neutral? If you replaced the transmission, there could be a shift linkage adjustment issue or even a faulty neutral/park safety switch.
Let me know. Also, please confirm if this is a 3.0L V6 or a 1.8L 4-cylinder. Additionally, if it is a Quattro, it should have a 6-speed automatic. If it is the 4=cylinder, it likely has a CVT. Please let me know which you have.
See pic below. I included the starter circuit so you have a reference.
Image (Click to make bigger)
Wednesday, February 15th, 2023 AT 7:31 PM