Glessed, I went through all these steps you are going through and came up with many "reasons" for the problems on my mower, but it turned out to be the lightweight plastic needle valve shown below. This is a KNOWN B&S problem. When I was first posting about this problem, another poster told me about the float problem, but I just didn't pay attention. Only when I ordered a carb rebuild kit did I notice the needle valve in the kit was brass instead of lightweight plastic. Replacing the needle valve took care of the problem permanently.
Like you, I would remove and clean the carb and it would work for a week or two and then quit again about 20 seconds after starting. I'd disassemble the carb and it would all be good again. After doing this three or four times, I figured out it was the float needle valve and started using compressed air (very low pressure) on the carb fuel inlet hose to blow open the needle valve. Finally, this practice became tiresome. I took apart the carb for one last time and replaced the needle valve. END OF PROBLEM!
So please, if you have a lightweight plastic needle valve like in the photo below, replace it with a brass needle valve. I think you'll be amazed at how quickly this problem disappears.