This is what I ended up building. It is lots of fun to ride! The deck is made of sprocket gears and sections of wheel rim, and the rear dropouts hold a wheel from a kids' bike.
I watched for discarded mountain bikes with sturdy frames, and picked up a bike with partially rectangular tubes and front shocks, but no front wheel. I pulled a decent wheel from another bike and swapped in a set of cruiser handlebars.
I use a cutting disc in an angle grinder to cut bikes apart. To start, I separated the top tube from the head tube, and the down tube from the bottom bracket. That left the pieces seen above: the front shocks, head tube, and angled down tube, and the rest of the bike.
The plan was to wrap the down tube around the wheel so that it is ultimately plumb, straight up and down. I opted to cut out wedges and cold-bend the tubing before welding the bends shut again.
In the photo above, I've marked the wedge that I intend to remove with soapstone.
After cutting out the wedge, I was able to bend the remaining tube like a hinge, closing the wedge to a weldable line.
Here's a close-up.
I removed a similar wedge at the second bend in the down tube, so that the last segment will be plumb. Then I removed the finish in the area of the welds and tack-welded the bends closed.
The plumb section of down tube did not come nearly close enough to the ground to support the deck, so the next step was to find a way to extend it (and hold the rest of the scooter up).