Clean the carb properly first . When you open the throttle the engine is most probably getting too little fuel and too much air flow - therefore the engine stalls. The carb jets / channels are probably partly blocked.
What happens if you press the starter button without pulling the brakes ? If the starter does not swing the engine while you are not pulling the brakes your scooter - that is the way the system is suppose to work. I think in such case it is just co-incidence that the engine started when you released the brakes.
Putting it in another way - The only purpose of pulling the brakes is to complete the electric circuit between the battery, ignition switch, starter button, starter solenoid, and starter motor itself. If you don’t pull the brakes, that electric circuit is not complete and the starter motor should not swing when you press the button
