17hp 1 cylinder B&S on 1998 Scotts mower runs about five seconds and dies

I bought this little used, but sadly neglected, mower at a garage sale last week. At least they'd kept it inside. It looks new. I figured I could make it run, but so far I'm not having much luck. It had no old fuel, so the tank was clean. I began by removing the Welbro carb and cleaning all its orifices with carburetor cleaner and a strand of wire. (Twice now) The needle and seat are functioning properly. The anti-diesel solenoid was sticky, but freed up nicely with the carb cleaner. It actuates when the key is turned on now, or pushes in by hand. It didn't before. There is good gravity fuel flow to the carb. The choke is fully closed with the throttle on full-fast. The choke did need to be adjusted. The governor spring is hooked up right. I think it still has a fuel starvation issue. It will run for a few seconds. If I let it sit for a minute or so with the key on, it will fire up again, and run for a few more seconds.

Any ideas?

17hp 1 cylinder B&S on 1998 Scotts mower runs about five seconds and dies