You can check if the brake switch work by pushing the brake pedal, if it work, the brake light will be on. If the brake light don't come on, it mean you have a bad switch OR the light is burned. Depending on how your wiring is done on your machine, if one or the other do not work, you won't be able to start your machine.
You can also unplug your battery, positive and negative, and do a continuity test between the ground from the solenoid and the ground you remove from the battery. From what you say, when you push the brake pedal, it should ground the solenoid to the ground from the battery. But you might also have a relay somewhere under the dash that need power to ground the brake switch.
The biggest problem is, every machine wiring is slightly different depending on the manufacturer, without doing all test, it's hard to say how your wiring is working now.
In the video, link under, I do not include pushing the brake pedal to start your machine. Some machine came with that protection and some didn't. It depend on the driver, if you have young kid driving, this is a good thing to keep but if only adult drive it, not so much.
I did a video on YouTube for the basic wiring.https://youtu.be/4U1ZulC22Zc