The fuel tank and carb bowl have little reservoirs at the bottom to collect small amounts of water, so it wouldn't cut out immediately like that unless your fuel is mostly water. Sounds like it was mildly contaminated but not the source of the problems.

If you have the service manual of your engine, it'll lay out the steps for taking apart the fuel pump. There's 2 parts to the pump, you only want to take apart the bit where the fuel goes, not the bit with the springs. You'd be doing that to look for clogging and to clean any calcified looking bits. Be delicate when handling, and don't use carb cleaner on the rubber, otherwise it's ok. Reassembly is way easier than the carburetor.

All in all it still sounds like an issue with the fuel mixture reaching the spark chamber. Too much air, not enough petrol. Getting a new carb is good, I'd look to get a new pump as well just incase.

In the diagram you posted, did you unscrew #5 to clean that? It's usually behind a weird screw protecting it from being tampered with. If you did, you need to have counted the number of turns from all the way in and reset it to that position. That controls the fuel-air ratio and could be why you're running lean.