I'd be tracking the wires and connections down. Something isn't telling the starter relay to activate. I doubt it's the switch if the brake light works. A simple test light or multi meter ought to be able to find where the problem is. Here's some youtubes on how to change and test the switch and it should show enough to know where to test the connections. Back probe the connections if you can. ...
https://www.youtube.com/results?sear...+scooter+brake