tug.cap, If seat safety switch test OK, could have either fuel delivery or ignition spark issue. If the fuel flow valve is off, the tank is empty or the fuel line/filter is clogged, fuel shut off solenoid faulty if equipped, faulty pump if equipped, dirty/damaged carburetor, gasoline will fail reach inside cylinder and engine will not start. A bad spark plug or failed ignition system/parts points, condenser,coils on older mowers, or faulty electronic ignition on newer mowers can result in weak/no spark not allow fuel to ignite. Below is a video to show testing seat switch and some trouble shooting links that may help in diagnosing your riders problem. Good luck.
I hope this helped you out, if so let me know by pressing the helpful button.
https://www.youtube.com/watch?v=a3dvvcpWgKA
http://www4.briggsandstratton.com/miscpdfs/RNT/Murray%20Electrical%20from%20MA-S-1612%202004%20Quick%20Reference.pdf
http://www.mtdparts.com/equipment/mtdparts/knowledge-center/maintenance/riding-mower-electrical-troubleshooting
https://www.briggsandstratton.com/na/en_us/support/faqs/browse/engine-problem-solving-tips.html
https://www.briggsandstratton.com/na/en_us/support/faqs/browse/ignition-system-theory-and-testing.html