Just forget about the rotating around sphere part, it was just to illustrate nearly-at-rest bodies exhibiting weird behavior. In my sphere part cases, it was rotation. In your case it's translation, causing issues. I made my example somewhat convoluted, I guess. 😅
Yeah, I had the hunch the animation which skips y-translation will most likely be unaffected.
Are you currently animating the position of the CharacterBody3D or the CollisionShape3D itself? Can you switch them out and take a look if it helps? (I know we are fiddling around here, Godot Physics doesn't have a good track record for pin-point accuracy (yet), nor is it documented when which part of the physics / node system triggers which reevalution).
Coming back to the two capsule experiment: Maybe I don't wrap my head around the multiple translations, but shouldn't one be able to keep the "feet" capsule static in the animation and do the rest by animating just the y translation around that? So that from the perspective of the feet sphere, we are not moving once we connect with the floor again, thus not retriggering the "sliding" evaluation?
One more option you have is using godot jolt physics, as a drop-in replacement. Since it's a specialized more mature physics solution. It might also be interesting to see if the issue persists, if you decide to open a bug report.
In regards to your MRP, I'm no big fan of doing this "here" in Discord. This is more a bug report thing, with a (git)-repository where people can clone from and commit changes to. Sending zip files around is to 90s for me, sorry. 😊 It will however be appreciated on the GitHub project (I guess).
One more thing, did you try the GoldSrc character controller (you said you tried most/all, but maybe don't know this one). It's fairly elaborate try to recreate Half-Life 1 character behavior.