The fact that some older vespa models had manual gears, proves that scooters can have manual gears.
Although I probably prefer a motorcycle style of gear mechanism, but one on the handlebars is not bad either.
Just to get away from the oh-so-common CVT.
I've seen the performance of this bike:

and athough the bike was based on the CF MOTO 244cc engine (or I could be wrong, the website calls it a 227cc engine), with 5 manual gears, it completely blew the BMS TXB 260 (with fuel injection) and CVT, out of the water; ( in acceleration that is; in top speed, the BMS was faster); despite the BMS having a better engine (based on the Linhai 256cc and fuel injection).
BTW, there's no better example of a scooter, than the Suzuki Burgman. It's a scooter, despite of having some small part of the engine (cylinder) inbetween the legs (has a horizontal placed cylinder).
The BMS260 is based on that Burgman.