Random no start, no crank

I have a ‘99 auto 3rz that will have a random no start no crank issue.

I have put a new battery, terminals, and checked to make sure they are tight. “New” starter contacts with probably 10k miles on them.

It will crank without hesitation until randomly it won’t. Tried putting in neutral and nothing. Go back a little later and it will start no problem. I tried shifting through gears and back into park when it won’t start, and does not seem to have an affect. I wait a few minutes or the next day and she will fire up.

HELP. Starter could be going out even with the replaced contacts? Neutral safety switch? Start throwing parts at it??

 

What is “new starter contacts with probraly 10k miles on them?” Battery cables good, not corroded? Cheapo battery terminals?

 

I rebuilt the starter. Good cables. OEM terminals.