Until you said your bindings were set as far apart as possible, I really could not have spotted it from your video. But have you measured your stance? You could be rocking something like 24 inches or more stance width if you are using the widest bindings settings. I seriously doubt you needed to use such a wide stance to obtain the extra stability you are craving for.
Your description of your board wiggle longitudinally and that you felt like about to catching an edge, could be because you were not pressing into each edge hard enough during your lapse of casual moments. As someone has said before here, if you are already committed on an edge, you shouldn't catch an edge. Were you close to flat basing when you experienced your instability?
There will be more people spotting what might be wrong with you as this thread moves on. What's really ironic is that I ride quite like you especially on the toe edge as I 'cruise' down. I might be leaning my body backwards just a tad more than you on the heel edge, not a lot more because I don't feel comfortable leaning too much into the blind spot ...... probably a natural instinct haha.
Fortunately I don't have all the instability issues you seem to have, even though there are other areas I want to improve on.
I agree with you regarding the wide stance. I wasnt/dont feel so comfortable with it but when i raised the subject with the instructor he mentioned that it was just a matter of getting used to it and that the extra width would help with the balance while sucrificing some maneuverability. Dont know about this topic so i guess i should leave it to the experts.
About instability...i cant really recall. Didnt focus on it and only thought about it when i started wiggling.