I've also tried to file flight plans with the alternate the same as the
departure, and they wouldn't take it. I didn't argue, and just gave
another nearby alternate (e.g. DET instead of PTK).

They have no basis to refuse to take it. The alternate airport doesn't
even get processed into the system beyond the originating FSS.

I know of an operator who operates a Gulfstream 5 on a regular basis from
LAX to HNL. During the winter when the weather is really churning in the
islands with a tropical convergence and all that brings, he files LAX as
his alternate. Experience has shown that operator that since they can
carry the fuel it is sometimes better to turn around should things be
nastier than forecast when they arrive in the area over there.