If it does not reliably start in neutral. I'd start by replacing the switch.

If I read that wrong and it's only a problem in park AND all the gears work correcting relating to the shift indicator.
It may be worn linkage. After years of use some of the pins wear, become smaller. Also the holes wear and become oval. So the linkage can not push the transmission fully into park. You adjust to compensate, then you can not go into 1st.

Bypassing is bad advice.