Sounds like you are on track with your diagnosis of neutral or start contacts in key switch. Check for voltage on the start lead when you key the start position (you will need an assistant unless you have very long test leads that will reach into the drivers position). If no voltage try jiggling the shift selector a bit while keying to start position. I would check for voltage at the park/neutral switch (key side of circuit or both if you are not sure) while keying to start position. Still no voltage? .... it is the key switch in all probability and that will be a pain to change out I expect. Maybe a discretely mounted dedicated start button is going to be much easier to do in that case.