Don't forget the spring!

[bgsengine]

I thought it was the fuel pump spring. The primer does the "choking"?

Umm yeah - it IS - to be technically correct in the modern age...

but as mentioned in prior posts - Briggs typoed on it or recycled part names from older parts books versions when it WAS a diaphragm choke spring - as I said - back when they had choke-a-matic carbs (Looks almost like the pulsa-prime externally.) They had the Vacu-Jet and the Pulsa-Jet carbs, and they had chokes that could be manually controlled , or semi-automatic, or fully automatic, and later, they even had thermostat springs to help open the choke plate.. It was the automatic choke that used these springs originally - and so their parts catalogs probably were just a "copy/paste" or the part name rather than re-naming it to what it actually is now - the pump spring.

[Merkava_4]

And here I was thought it was the fuel pump diaphragm spring.

I would have thought the same thing, but apparently the engine runs just fine without it, but the carb needs that spring to get the engine started when it's cold.

[bgsengine]

Ya got to understand how they work - when you just replace them and refill fuel tank you still need to get all the air out of the tubes and jets and fill the fuel well - primer just pulls fuel up and pushes it into the carb throat - (you can actually feel the difference between it pushing air and pushing gas, if you pay attention)
The spring works like the "push" side of the fuel pump- every intake stroke there's gonna be vacuum pulling up against the spring which pulls fuel up through the pickup tube and when the intake valve closes, vacuum is lost enough for the diaphragm to return and "push" fuel and dumps it into the fuel well for the main nozzle..
Once you have had it running for a while, it IS possible that the diaphragm can return all by itself (or during compression release event when a tiny bit of pressure makes it back to there.)
But for starting purposes after it has run out of fuel you have to move all the air out of the pickup tube, and fill up the fuel well completely full - The spring will be most important during the cranking and start-up phase to refill the fuel well.

So.. your description of the event with the spring being missing sounds about dead on to what one might *expect* to happen