DIAGNOSIS The problem isn't with the self-propel. The blade control bail return spring is weak or disconnected, so it's not keeping the bail engaged. The engine is set to stop when you let go of the bail.
FIX Inspect the blade control bail mechanism on the handle. There's a spring that returns the bail to the "run" position. If it's detached or broken, you need to reattach or replace it. You'll likely need to remove the handle cover to access it.
IF THAT DIDN'T WORK The cable connecting the bail to the engine might be frayed or stuck, preventing full engagement. Follow the cable from the handle down to the engine and check for kinks, damage, or disconnection.
HEADS UP Operating the mower by manually holding the bail forward is a temporary workaround at best and defeats the essential safety shut-off. Fix the return mechanism properly.