You are right in thinking these scooters were made for smaller Asian framed people, or kids. Especially when modding them, us bigger folks can overtax the stock hardware pretty quickly. To paraphrase what "they" say, for most scooter builds you can pick two from the list: fast, reliable, cheap.
First we need some more well defined goals.
Top speed: "goes currently around 15mph". What would you like to hit? That chassis is probably pretty squirrely at 25 and up. If you want to hit 30+, you should think about lengthening the frame and going disc brakes.
Run time: "runs out of steam with little riding time". This is where the size of the batteries comes in. The larger the battery AH capacity, the more runtime you get. When manufacturers say "up to 30 minutes of runtime" they mean on a flat, smooth surface with a 60 lb girl on it. When we bigger fellas get on, that number goes WAY down since the controller is pretty much constantly pulling max amps. If you kept your current motor and controller you could get more runtime by going with larger batteries, but top speed and torque would be the same.
Hill climbing: "wont pull much of any hill 10-20percent grade". This can be solved with either a gearing change for more torque (but less top speed), or a larger controller to feed more amps to the motor. I think the stock controller on that scooter is around 25 amps. Going with a 50A+ controller gets you more pulling power and acceleration, but also sends a LOT more heat into the motor. It is a very quick way to burn up a stock motor.
So if you wanted 25 MPH top speed, more torque for small hill climbing and acceleration, and longer runtime you are looking at a motor swap, new controller, and larger batteries.
I know Rob was selling a 1500W brushless motor that would likely be perfect, but it may already be sold. His recommendation of a yk43b controller (pushes somewhere between 100 and 150 amps, it is mean and cheap), an industrial motor (like the ones used on pumps and forklifts), and big 22AH batteries would definitely do it. Might be a bit of overkill, but we like overkill!
Here is a picture comparing a 350W Unite motor and a 1HP industrial, from my current scooter build:
Of course, there are tons of other options. Like going with a bigger Chinese Unite scooter motor, using RC car/plane motors (which can bet absurdly powerful for the size), etc.