Engine will start if starter contacts are shorted while ignition is in the on position. I checked battery and connections, fuses and relays on firewall, and replaced starter. Solenoid clicks once when ignition is turned on.
Try applying 12 volts to the small solenoid lead of the starter. If the starter works ok doing that then you have a problem with the ignition power lead to the solenoid. Possibly the neutral safety switch is the cause. See if it will start in neutral in that case. If it does work then the safety switch needs to be checked. If powering the solenoid doesn’t help then the solenoid is bad and you may as well replace the starter.