Just a guess, but it sounds like the engine is locked up....since you have replaced all the other stuff that usually takes a dump first. Follow my logic:
If the relay is energizing and the starter is trying to turn (the loud click you hear), and the bendix gear is engaging the engine flywheel, (I assume that the new starter had a new bendix), then the only thing left in that chain is a locked up engine that won't turn.
Suggestions before panicking... remove the starter but leave it connected to power. Make sure that the starter is actually engaging, spinning freely and that the bendix is throwing out like it's supposed to (keep you hands and fingers and limbs clear of the gears and momentarily press the start switch). If not, you might just have purchased a bad starter. It happens.... If all of that is working properly OFF of the engine, then try to turn the engine over manually (use a pry-bar to move the flywheel). If it moves freely, then the engine's NOT locked up. You could have caused the bendix gear to bind when you installed the new starter.
Just some random thoughts that might help.....
1999 Holiday Rambler Endeavor, 36' Gasser
Triton V10, Ford F53 Chassis
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