Briggs will only start, and run for a moment, if I manually close the carb butterfly

Briggs 12H802-1767-E1.
 
Two things to double check.

1. Fuel bowl retaining bolt/main jet is clean. The main jet is in the center of this bolt that goes down to a cross hole.
2. Since this a primer type carburetor, that primer is actually priming the carburetor. The air filter/primer are bad to warp leading to non-priming. Requires either a new air filter base or two gasket glued together and beveled to ensure sealing of the primer air pulse passage.

It is right the opposite on governor operations. The spring holds the throttle vane side open when not running and the governor will work against spring tension to close the throttle vane to slow the engine when the engine is running. Governor are meant to slow the engine. It is the delicate balance between this governor closing action and the spring wanting to open the throttle fully. Without the spring the engine will never do anything other than idle.
 
It just will not prime the carburetor without it in place as priming takes the place of choking the engine.
 
As the new air cleaner gasket take off the cleaner and look at the impression of the air cleaner makes on the gasket. It should be uniform. And don't over tighten the mounting screws as this will warp the air filter base.

Update:
 
I went to the auto parts store to get some more carb cleaner and went ahead and bought some starting fluid spray.  A quick spray in the carb and it started just fine.  I guess the primer bulb is cold and stiff and the mower was just too cold.
 
Runs fine!

Interesting...just looked it up!

Would it be safe to shoot a squirt of carb cleaner in there instead?

This I did not know Roy. Thank you. No starting fuel. Real fuel. Thanks!!