Your engine has the Pulsa-Jet carburetor, which is both the best and the most complicated carburetor Briggs had in those days. Here is Briggs' check list of possible faults to be considered in your situation:
1. Leaking carburetor mounting gaskets
2. Gummy or dirty screen or check valve (Pulsa-Jet and Vacu-Jet carburetors)
3. Inlet valve stuck shut (Flo-Jet carburetors)
4. Inoperative pump (PulsaJet carburetors)
You can ignore Item 3, since you have a Pulsa-Jet not a Flo-Jet. Hence there are three probable causes of the problem. The first one is fairly easy to check. You may recall that I suggested earlier that you unbolt the carburetor, air filter and fuel tank from the engine to get access to the valve cover. The two carburetor mounting bolts are the two I showed circled in red on a sketch of the engine cylinder. To deal with possibility 1, you need to check that both of those bolts are done up firmly, not rattling loose. It is not all that likely a cause for your problem, but it is very easy to check. If it isn't that, you are going to have to remove the air cleaner, carburetor and fuel tank from the engine, so it makes sense to check it first.
The picture shows an 80202 about the same vintage as yours, but they are not all the same, so first, post a picture of your air filter, then remove it from the engine. For the one in the picture, you just unscrew the large screw in the center of the top of the air filter (circled in yellow), then you can lift off the entire air filter from the carburetor. If the filter is substantially different (e.g. a later one with a paper filter) you may have to remove the top cover to access the mounting screws. Post a picture of the carburetor (immediately beneath the air filter housing) when you have done this, so we can discuss the following steps.