okay from the diagram it appears you have a model 12 quantum engine,
Now, lets get some terminology straight, You said it has a primer, but then mention a choke door (choke plate), on these briggs, you either have a Primer and no choke, or a Choke and no primer.
So does your carb look like this one
View attachment 58908 Or this one,
View attachment 58909 with a throttle shaft and choke shaft?
Now, the problem could lie within the amazon carburetor, as it is not uncommon for them to sell Duds from the factory at a discounted price.
The diagram shows it originally as having a Primer/ no choke carburetor.
Anyway, to the meat and potatoes of the problem,
It is running too lean to start properly (lean mean not enough fuel), whenever you press the primer bulb, you should see fuel squirt up out of the main jet.
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