Follow-up... The hinge pins and bushings did take the play out of the door but the door still required a strong slam to close. A Dorman striker bolt installed today did the trick. It probably needed both hinge pins/bushings and the striker bolt. The bolt looked good - the thin plastic membrane did have some cuts in it which must have made the door so hard to shut correctly. Slamming the door hard enough to close it correctly would have done damage to the window tracks, latch, window motor, or mirror workings.
Thanks again for the suggestions!
Thanks again for the suggestions!