We've been trying to hammer the solution to this issue to our customers since B&S & many other manufacturers, switched to their aerodynamically shaped fuel tanks for esthetic-reasons, mainly to match the new OHV engine styles. In doing so they knowingly created a nightmare for owners of same equipment.
These tanks have a domed-like top, & various baffles & wells molded into the bottoms of the tanks. These features, combined with, ANY moisture AT ALL, in the air, WILL create the issue the OP is dealing with.
Owners believe & insist they never use old fuel, or, "it's new fuel," "they just checked/cleaned the carb," etc., etc., etc.. It doesn't matter how well you care for your fuel. The one solution is to ALWAYS keep the tank as full as possible, but this does not assure a total fix.
Engineers assumed their fancy tanks would be the wave of the future. Together the domes & the wells create the perfect "condensate creator + water storage system" which, once the wells are full of water, begins the nightmare.
1) Condensation collects inside the tank in the domes.
2) Vibrations and equipment movement shake it into the fuel.
3) Inherent liquid densities cause the water to sink into the wells. (One of which is also the location of the fuel line to the carb.)
4) In average to high humidity regions, it's typically about one year before the problem shows itself. Dryer climates may never see it occur.
5) Consumers often go in & drain the float bowl, (correctly so) but neglect to clean out the tank. Worse, since the tanks have several of these wells in them, you can't just pull the line and drain it, for the other wells will hold their built up water. (We've occasionally drained them, and blotted up the visible water with a clean rag attached to some sort of poker.)
6) Temporal fixes (draining the bowl & using the equipment until the bowl refills with water again) are responsible for a secondary stage of the problem. e.g. Eventually, the jet/pickup tube that the carb float bowl plug screws into, will become corroded & or, partially plugged. Oft times the actual jet is in this plug, thus, leaning out the fuel to the engine & creating the, "runs when you prime it" example of the OP.
You can use a tiny wire to gently scrape away this corrosion in the plug/jet, as well. Those carbs where the jet is in the pickup tube screwed into the carb body, you'll have to remove the carb & use a similar wire to clean out the tube & its tiny orifices inside the carb body. If the bowl was punched out of Chinese steel, it'll rapidly exhibit un-repairable corroded-thru leak point in the bowl. (new one, or try JB weld.)
Nonetheless, trust me when I tell you, (post-multiple thousands of dollars to do this semi-simple work for customers) these systems need routine cleaning.
Also, contrary to popularly offered wisdom, you never want to leave these "drained" in the off season. You should fill them to the brim, and if possible, start them once or twice in the off season, & cap off the tank again.
I (could) offer names of B&S employees who've told me, this was a KNOWN design flaw, and yet, they've left it be for decades in some cases. One of our habitually condensation producing machines, I finally grew tired of and ripped the plastic tank off and installed an old steel tank on it. After several years, never had another condensation problem again.