What you were describing, I think this has been done once or twice, but I can't remember where. There have also been projects like the Straight Razer, http://www.evalbum.com/4020
, which could be further downsized.
Downside is that whatever you do you would probably want to use LiPo, probably, but because you're dealing with flight I think you want the LiPo *really* well protected. Unless, maybe you can get your hands on 2.3ah a123 or small headway (sidenote: You might not need a BMS for the a123...I know that on a lot of projects by the guy I linked, he just uses balance wires and balances every rare blue moon).
In picking the scooter you get I think there are a couple tricky problems. The lighter alluminum ones are great, but tend to have very little clearance to stick lipo under the deck so maybe you would have to either somehow extent the wheels or make a false deck above the deck, to fit the batteries. If you keep your range desires low I guess that makes this more feasible.
The scooters designed for E-vehicles (like the electric razor series) have tons of space (they're made for SLA), but they are usually made of super clunky welded steel. Definitively not cool.
If you work on this, please document at least a little about it ^_^. I'm a bit curious how it goes.
P.s, the constant top speed no matter diameter of wheel makes sense because your motor is spinning a wheel (or it's drum) as some rotational velocity. That, times the radius of the friction driver, is exactly how fast you're going. The wheel size is not doing any reduction or gearing at all.
New to endless-sphere? Notice a lot of signatures recommending Ebikes.ca? That's because it's the best place for a quality and manageable, (but still 'real'), first build. Justin is a solid supporter of electric bike development, and a good source of equipment.