Since your starter is brand new and was bench tested before install it has to be a wiring/ connection issue or battery issue.

The relay that the PO added was probably to circumvent a weak/ dirty ignition switch or NSS. The starter solenoid draws 8 amps and goes directly through the ign and NSS switch. A weak/ dirty switch will not carry 8 amps so it was rewired to turn on a relay instead. A relay coil needs much less current than 8 amps to pull in. needs around 100 ma depending on the relay.

Don't forget to check the ground strap already mentioned.

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85 300D turbo pristine w 157k when purchased 161K now
83 300 D turbo 297K runs great. SOLD!
83 240D 4 spd manual- parted out then junked