Hi Rob,
It doesn't surprise me that you prefer up to down. I think most riders, even experienced and confident riders prefer the up hills to the down hills and there is definitely an adjustment to make between the two in our approach braking, throttle and trail braking, and exit.
I like your comparison between cage steering wheels and bike handlebars. As you and others have pointed out, stiff arms add to the fatigue and anticipation even when in the straights and especially through the turns. And I would agree, many of the sportbike drivers I ride with, see on the streets, and even some on the track, rely on the handlebars for their upper body support. I can't ever remember doing that, but I'm sure I did when I started riding, but I started riding about 55 years ago, and they say that is the first thing to go!!
Most of the reason riders lean on the bars, I think, is because of poor body position and especially their backs. When I see a rider arch their backs, upright or back, almost certainly they have weight on the bars. Another giveaway sign is every 30 miles or so they are releasing the throttle grip and shaking their hand to relieve the cramps. And of course there is the release the left bar, left hand on their leg so they can get a more upright position to their back to relieve the aches!
And most look really apprehensive and stiff going through the turns.
There should be no body weight on the bars at all, none. We should be able to take both our hands off the grips by an inch and feel no necessity to flex our leg, butt, and back muscles to hold our riding position. If you try that, you are probably going to have to flex some muscles to keep your chest/head from falling forward. If we do, then we know we need to work on those massive muscles in our calves, thighs, butt, and lower back, while relaxing our mid-to-upper back, shoulders, and arms, using only the light required hand grip on the handlebars. That is, just enough grip to assure we have control for the specific road conditions and maneuvering such as in the turns. In normal riding on good roads, it takes no pressure in our grip, just a little feel to attain the required counter-steer for the direction we want to go.
Once we can let go of both grips and retain full support of our body riding position with out mid-chest down muscles, we need to enhance that control by simultaneously shifting our body weight allowing our butt to slide on the seat smoothly without upsetting the bike dynamics. If we do it right, and are new to riding, our legs should hurt after a ride, rather than our back, arms, wrist and hands. Outside of things like a tank slapper, etc., our feet, calves, thighs, butt, and lower back control our riding position, so our upper back, shoulders, arms and hands can control the bike to go where we want it to go with small relaxed pressure adjustments. Please know what I say isn't expert advice, or any kind of advice, it's just the way I ride and seems to fit me well.
At my age I should be riding a Harley bagger like many of the seniors I ride with, but even the sit-up position of a Sport-touring bike makes me feel even older than I am! I need to arch my back forward over the tank and hang off the side when my speed dictates it (and it often does!) in a turn. So when I'm cruising along in around triple digits, my arms aren't straight, they bend at the elbow and my hands are lightly laying on the grips.
I don't think it will be very long before you start feeling better about your left turns, and soon after that you will most likely forget their ever was a difference.