Starter keeps starting!...

With new wiring, I would not be removing the starter just yet.

This is extremely easy to test/isolate the problem. Remove the starter sol wire at the relay. Jumper that to bat + terminal & remove after a couple of seconds. If starter releases, starter is ok. If starter continues to run, starter or mesh is faulty.
I don't know if you read all this but I've tested everything and it's the starter..
Again reason being I don't want to do a lot of starter testing with it on the engine is the engine has never been ran... I'm not going to try and slowly break in the cam trying to test a Speedmaster Chinese starter...