This brings up the good point that "balance" can mean one of 2 different things:
a) Muscular coordination required to remain upright.
b) Sense of balance within the inner ear.

Two very different things.

I think I'm definitely category A.

The feeling of lurching toward doom on drops is the biggest issue, if I had to pick one.

The last time I did that was when I was about 15. Doing a long gentle curve. Wheel spun around in front of me. Over the bars I went. Bent rim. Walk of shame back home.

I walk my bikes mostly by the seat, but have never tried riding no hands since then. Not worth it for me.

I don't remember which bike it was on. I'm thinking it may have been the Viscount (Stolen Freshman year HS), as I don't think I lost a rim on the Colnago until college.

Not at all. When I rode motos, you kept your 'focus' on the point where the centerline and the fog line came together. If you're looking at what's right in front of your wheel, it's too late to do anything about it. You look for the hazards out there, and let your peripheral vision handle it when it gets closer.

Stability also has a lot to do with how well 'tuned' your supporting muscle groups are. Things like hip flexors, obliques and lats. These support your trunk above your waist, and the better conditioned they are, the less you rely on your arms for support.
I found that Tai-Chi was excellent in this regard, with it's emphasis on moving smoothly from one position to the other; it stresses supporting your body by the muscles, rather than the skeleton. You can accommodate the bike moving around under you, rather than reacting to it. Because when you react to the bike, it reacts to you.