The fuel pump should give a steady stream of fuel. It wont spray aggressively as it's may only be 2 or 3 psi but it should give a decent volume of fuel. How about one more way to prove where the problem is? Could you connect a qt size bottle of fuel straight to the carb and squeeze feed it? If it stops surging, you have a supply issue if it doesn't change, go through the governor adjustment.
I dont know this engine but be sure to do it correctly. Small engines often use a centrifugal gear drive governor and if the internal linkage is turned too far the wrong way it can cause catastrophic damage. First be sure the linkage is in good condition. If the linkage and holes that they ride in are worn and there is too much slop it can cause surging or hunting. Another trick to prove if it's actually lean is to falsely enrichen by figuring out how to feed it just a very small amount of extra fuel as it surges.
You are running it with the air filter when it's surging aren't you? It needs the filter in the system as in theory it would give it more air (lean) without it. Some engines are more effected by this than others.