I have a similar engine on a Craftsman Pressure washer. On mine it is labeled Quattro 4.5 HP. Same air filter housing and carb as yours. I think Generac made it. Here is what I did to get mine running at 100%
Replaced the head gasket. The head comes loose over time. At least re-torque it to spec.
Did a compression test on it after that, and it only had 38 PSI. Adjusted the valves and that came up to 70 PSI. That is within spec for my engine. Valve guides are worn on mine from all the hours but it still runs well.
Replaced the carburetor with a B&S original, correct number.
Filed the top of the fuel tank level as they can warp. Some shops use wet and dry sand paper on a piece of glass. I was a machinist for some time by trade so using a file to do that was quicker and no need to clean the tank afterwards other than compressed air. Don't use a file unless really good at it because one small slip or getting the angle wrong can ruin the tank.
The fact your primer does not return likely means there is some kind of blockage like at the screen like Bert said.
When replacing the carb, make sure the gaskets go in the correct order. Do not over-tighten
Did a standard tune up, plug, oil, new air filter (I found that my engine ran better after taking the old air filter off even though it wasn't all that old. New filter solved that. I also set the air gap because I removed the coil but that is not needed.
Use canned fuel, the washer just runs better with it. Starts easy and has no surge or stumble going from slow speed to high speed or when the wand is pulled. Starts like when it was new.