I did it all the time when I was a kid but lost my nerve when I broke my collarbone, it took me a while to get it back. I'm fine now. Put the bike in a higher gear, you want to be mashing. This lets you put more control into the bike with your lower body. I don't know about racy vs. not-racy geometry. My Paramount is definitely twitchier than my Hardrock but both tend to find the center. But I had a weird experience on a rental city bike a few months ago, it just wanted to keep leaning over further.

A motorcycle with no hands is pretty weird, they are VERY stable compared to bicycles, they want to stay straight and take a lot of weight transfer to steer.