Here's my guess. It's a self-priming model with the float style carb. At the bottom of the carb float bowl is a bolt. Take the bolt out, it probably has some debris stuck in the small hole down by the base. The bolt is hollow and drilled through, but since there is no inline filter, debris gets stuck in the bolt, that stops the flow of fuel to the carb venture. I've fixed a bunch of them that had that problem. Want a permanent fix, add a little inline filter from the tank line to the carb, something from napa, autozone, o'reillys.
ETA: Didn't read your post thoroughly. If it has the older diaphragm style carb, the thing that goes bad most often is the diaphragm, which is also the fuel pump. No matter the make or model, removing the carb and spraying all openings, jets, etc out with carb cleaner can't hurt and might be all it needs.