I've been taking pictures on the fly off my motorcycles since the 70s. The first camera I started doing this with even had a manual film advance, no digital, no power, no batteries.....
For me what works is keeping my camera on a lanyard around my neck. When I see something I want a photo of I simply grab the camera and point and shoot. I don't bother to look at what I'm pointing at so no taking my eyes off the road. When shooting scenery and things along the way while riding pointing in the general direction is really all you need to do. It's amazing what you can get from doing this. Most shots I take in my ride reports are done on the fly because like Jim said it wouldn't get done otherwise. I started doing this for exactly the reason you mentioned, no shoulders where you want to grab a shot and I wanted those shots.
I'm right handed and used to do this using a throttle lock but about 6 or 7 years back I started riding bikes without throttle locks so I tried doing this with my left hand. I mastered pointing and shooting with my left hand very rapidly within about 5 shots. I also discovered using my left hand for the camera and keeping my dominate hand on the bars it felt like I was in more control that using my left hand on the bars.
Now about the lanyard around the neck: Some people fear wrecking with a cord around the neck and a camera hanging off it. I have wrecked plenty (dirt) with the camera hanging there and it has never got caught on anything or caused any problem. It just kind of flops around along with me.
I have more fear at the thought of having a camera in a jacket pocket, held in place so if you fall on it your are hitting something hard, like falling on a rock.
The down side of having a camera hanging out there is in cold weather the battery will lose power sooner. Watch for dust and bugs on your lens.
Like everything taking photos on the fly can be dangerous or you can make it as safe as anything else. Common sense should dictate when it's safe and when it's not.
Keep taking those photos and sharing them with the rest of us.