Ditto much of what has been said. I have a climber's body and climbing has always been my strength. I have nine years on a V20 (now V20c), and I climb just about as fast on it as I do on my Specialized Aethos road bike (my local club is hosting its annual 12 mile uphill time trial this Sunday, and I am having a hard time deciding on whether to use the V or the Aethos; they're that close). I run a 36x30 max climbing gear on both bikes, 160mm cranks on the V and 165mm on the Aethos. The Aethos is about seven pounds lighter, and does have advantage when the gradients are steeper (above 8 or 9%). On 5% or lower, I think I start getting an aero advantage on the V that makes it the faster climber. Where the V really shines for me is with fatigue. I can climb a lot longer and farther on the V before the legs start to feel it. I don't ride with any data, so I couldn't tell you any numbers to compare the two kinds of bikes.
The wobbles and the anxiety that comes with it went away after enough time, mileage, and experience. It took a while, but with practice my technique improved and climbing on the V came to feel almost as natural as on an upright bike. I am a little more vigilant on the V because I have to control both hands and feet, rather than just hands, but I can hold a straight line close to the edge of the road easily enough. Tension works against this, but being able to relax takes time and practice; it's a V20 catch-22, but one that tends to improve for those who push through the learning curve. Gradients above 15% are challenging, but that is partly due to my largest cog being a 30t rather than, say, a 36t. But I don't see such slopes often enough to justify having that low of a gear. Clipless pedals definitely help by connecting you more to the bike. I use Shimano XTR SPDs because they have double-sided entry and the cleats are easy to walk in.
Two thumbs up for the Varia! Hear the signal on the head unit, check the mirror to see if you have a sedan or a logging truck coming, prepare as needed without surprise and keep riding.
Lastly, climbing is a personal experience. One rider's approach may not be the same or even work for another rider. The V perhaps magnifies this even more. My crank length or gearing selection that took me a while and some trial and error to settle on might not be the best for another person. But as long as the bike is properly fitted and the gearing is basically appropriate for the terrain and the legs pushing the pedals, you should eventually find the V to be a capable climber after enough time and practice.
(And the downhills are so, so much fun!)