Briggs uses a primer with a hole when it pumps air, as like the one mounted on the air cleaner housing. the no hole primer is used when the primer is part of the carb itself. Yes the carb bowl vents through the primer hole on the engines with the primer mounted on the air cleaner.
It is very common for the air cleaner backing plate, to warp were it mounts to the carb, and if it doesn't seal, when not prime. the fix is to double gasket the air cleaner backing plate to the carb.