I think the reality is paying anyone to do any type of engine rebuild on an older scoot will cost more than the value of the scoot. If that's your only option I'd probably just sell 'as is', for parts.
Before giving up on it I'd first want to see if the engine will even turn over. ...and not with the starter button but by putting a socket and wrench on the variator nut. Will the engine spin over? If so, you're in better shape.
Have you visually scanned the motor for any cracks, holes, major leaks, etc.? If it is truly gone it would be much easier to swap in a used motor, assuming any of those are available...not many used scooter motors available around me.
Having just completely stripped down and then rebuilt one of these (S200, air cooled, 163cc) I can tell you that it is actually a very easy process if you go that route. The things you'll need (other than any new parts) are a full gasket set, 2 crank seals ($60, total?), a flywheel puller and starter clutch nut removal tool (another $30?). There's an endless supply of Youtube videos showing how to disassemble and rebuild a GY6 motor.
So I'd first want to know if the engine was viable..if not, sell for parts and move on, swap in a used one or last option, rebuild the one in there, yourself.