Re: Engine won't start anymore.
The little filter looks great.
The shift interrupter is what allows you to get the drive out of gear. It grounds the ignition for a fraction of a second while shifting out of gear to kill the torque from the engine so the dog clutch can separate. A problem with that switch would not allow your engine to run, or you would not be able to get the drive out of gear with the engine running.
Before you tear the carb apart you should at least look at the little filter. It just takes a minute.
Another possibility is your fuel pump might not be putting out enough pressure.
Or maybe your choke is broken or way out of adjustment... That doesn't usually cause backfiring though.
The little filter looks great.