Alright, now I'm certain where the bug creeps in.
I took the same assembly of parts and did an animation of them, and published them - and there were no transparency problems in the animation or in the published AVI file. But this time I did one key thing differently. Before, in my problem animations, I started with most of the components invisible, and then brought them each into visibility, one at a time. But in doing so, I would alter the fade-in, by extending it to perhaps one second, as opposed to allowing a quick off-to-on. Making that brief extended fade-in has apparently been the trigger for my transparency problems.
In this new animation, I didn't do any extended fade-in's in making the components visible(the "duration" of each was the default 0.5 seconds). This makes the animation less pleasing to look at; with all the suddenly appearing parts. But there were no transparency problems.
I then took the no-problem file, and made a copy of it by doing a save-as. I then changed the duration of several of the parts, from the default 0.5 seconds to 1.0 seconds. Just by doing that, I instantly had transparency problems in the animation and the publishing. I made another save-as copy from the no-problem file, and tried changing the duration of the fade-in's from 0.5 seconds to 0.7 seconds. Again, I came up with transparency problems.
I then took the failed animation, with the fade-in durations set to 0.7 seconds, and set them all back to 0.5 seconds. It didn't matter - I still had transparency problems. Once you change the visibility duration of any component, everything stays messed up, and apparently can't be set right.
I don't know why this is happening now - especially since it didn't happen to me in the past with animations - and I constantly used extended fade-in's. But that's the problem I'm seeing now - and I sure would like the Fusion 360 team to fix this bug.