I had a very similar problem with my TO-20. It sounds a lot more like a fuel problem than an electrical one to me. I did try and clean my carb but this was unsuccessful. I don't think soaking the carb will help either if the problem is some non-soluble piece of crud that has blocked on of the jets. Since you know there was crud in the tank etc, I think it very likely that a small piece of debris has lodged itself in one of the passages or jets that supplies fuel. When you clean out the carb however, you should try and make sure all the intermal passages are free. I used a squirt bottle with some carb cleaner in it and pushed the nozzle into the opening and squeezed the fluid in. Some people using fishing line to push down into the passages (but do not use wire as it can damage things). In my case, the fluid shot out of the bottom of the passages freely. But this did not solve my problem. Anyway, unscrew those brass jets and give it a try. I ended up buying a new carb from this site (approx $200) and that solved the problem. All the other options, like a manifold lead, or an electrical problem don't seem very consistent with the facts in your case...like the choke helping etc.
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