OK, but even with BIN, do I lather it on everything, including the trim and doors which will be semi gloss?

Are you painting everything? Then it all gets scraped, sanded, tacked off, holes filled and sanded, and only then primed. You don’t glob it on, ever. And never without doing the rest of the prep first.

Like I said in the original post, everything has had joint compound put on it then sanded down to minimize imperfections. After the primer, the trim and crown will have a different color paint then the walls but all of them will have primer on them...

Did you scrape and sand before you filled? If not, you may have adhesion issues. Raw compound always needs to be spot primed first before the main primer coat goes on. You’ll probably need to do more fill and sand after the first primer coat shows up the issues as well. That gets block sanded and spot primed again. A great job takes a lot of time and a lot of rinse and repeat steps.

Yup, everything has been made as smooth as possible, and after the sanding, I rub down the wall with a damp cloth. This is more about the next step, using primer, and if it can be used universally on every piece. I will go ahead with BIN or the Sherwin Williams Shellac Primer

It's perfectly fine to use on everything.