What you've described should work. Only other thing I would do is to hit any really shiny surfaces with some sand paper to lightly rough it up and remove any rough edges, but like you, no way would I strip the woood!

You can ask about using the SW primer on a shiny finishes on the Paint forum for more input. Not sure how active it is over there.

I'll be curious to see what other ideas and comments you get. I've done all the painting in all of our homes, so I am a fairly experienced 'non-professional" and I think your approach sounds very reasonable. I have not ever used the SW Adhesion Primer though. But you have, and on fiberglass, so that seems like pretty trustworthy experience.

Regarding testing the product in a closet. Honestly, I think that's a waste of time. It actually takes weeks for paint products to fully bond and set up. You should plan to be extra careful the first month.

Be sure to follow the suggested wait times on the primer can and on the 'between coats' paint can. I think this is where people mess up. Too little wait time and too long can both cause problems.

Also, use TSP or Dirtex for the cleaning. It won't leave a residue, and that's important.

One step you didn't mention was caulking. Took me many years to learn this, but it is the trick to give you professional results around woodwork. One of those things that seems intimidating but isn't. Try Youtube for some easy demos.

The paint forum can be incredibly slow if you're looking for color advice (better here) or if you have a poorly worded question (less patience for figuring out what on earth you're trying to ask). However, many of the regulars know their stuff, so what's lacking in quantity is compensated in quality. Paintguy gave great advice.

Here is a link that might be useful: cearbhaill's question cross-posted in paint

TSP is also a deglosser so the need for sanding sould be very minimal.

You do NOT need to sand to bare wood, but there is no substitute for proper prep work that includes a bit of sanding. You need to take one of the all in one tools and scrape off any loose finish after cleaning and then scuff sand, especially the transition areas between where the finish has come off and where it is still adhered. Your goal is to make sure that any finish that is left on the wood is holding tight and that there won't be noticable bumps and lumps between the different levels where the finish is still applied and where it is absent. If you don't address those issues, any paint that you do apply can simply flake off when the underlying finish that you are attempting to cover ceases to adhere. And, you will not have a smooth nice looking paint job when the surface underneath has areas that have flaked off. They will telegraph through unless the transition area is sanded smooth.

Thank you all for your comments.
I have been persuaded to "scuff sand" and use BIN primer.
Waiting between coats won't be an issue for me as I am patient and have several rooms to do.

And FWIW the adhesion primer did a fabulous job on my fiberglass exterior doors- it's years later and they look just as good as they did the day I finished them. I was scared to death the first night we closed them completely- I had been leaving them open a tad to cure thoroughly.
Must have been two weeks before I had the nerve to do it!

You might look into BM Stix...I was talking to the paint guys about painting our garage door and he recommended we use that first....supposed to be better than BIN, but I'm not sure about your application...you should check at the paint store.

I love BIN primer, but would not use it to promote adhesion. There are products made especially for adhesion and that is what I'd use.