3 pin neutral saftey switch check?

It is difficult to check a neutral switch per se. They push in some distance for neutral and push in a different distance for backup lights. Some guys are finding that modern replacements "are not the same" and require fiddling with gasket thickness to get them right.

One old trick is to hold the key in "start" and with your other hand, work the shifter from P to N and back, and lsiten for a click or "try" to crank.

Shift linkage can be out of adjustment. The connector at the switch can be bad, corroded, loose, etc.