>There is no natural scenario when one's eyes would diverge

I like to look out a window at something distant, like a tree or car outside, then adjust my focus on the window pane itself (perhaps the little flecks of dust on it, or better yet one dot of paint on it).

That is a very natural scenario, and at least for me. super easy. I've done it for decades though, so maybe it's just natural for me, lol.

I can FEEL my eyes slightly squeezing to change shape (and thus change focus), and at this point it's simple muscle control, and it's fun.

Looking at something behind the screen (if transparent) or reflected on the screen (if slightly reflective) should do the trick.

This only gets you there if the physical distance between the stereo images on the display is <= the distance between the eyes. If it is greater, you would need to look at something more than infinitely away / behind you (your eyes would have to diverge). In particular if the screen is already quite far away (e.g. a few meters), there's not much more far away you can focus anyway, since there's not much difference in angle between a few meters and infinity.

I have to admit I don't understand your argument, but in my experience:

Stereograms are NOT meant to be enjoyed from across a room.

It's REALLY hard to 'catch' a stereogram from far away, so why are you even trying?

However, the "focusing on-or-through a transparent thing" that WhiteSage mentions works well at slightly further distances and can be used to help you figure this stuff out.