Mark, as far as I am aware, boiling water with detergent in it will have little if any effect on the petrol gum that is the key ingredient in most carburetor fouling. There are two ways to clean a carburetor. One is a combination of metal probes and compressed air (you apply the probe, then blow out the gunk, then apply the probe again, to each drilled passage in the carburetor). The other method is a spray can of carburetor cleaner. The carburetor cleaner is faster and more effective, but the stuff is unfriendly to polymers so it may shorten the life of plastic or pseudo-rubber parts of the carburetor. It must be blown out with compressed air, to minimise the amount left in there to do harm. Of course if you reassemble the carburetor, attach it, and run the engine right away, the petrol will tend to wash away the strong solvents from the cleaner.
Since you have a clear supply of fuel to the float bowl, the remaining issues are the main jet, the drilled passages, and the mixture screw. The usual way to attend to them is to consult a schematic of the carburetor then blow the cleaner through each passage. It is seldom necessary to remove the permanent plugs (usually Welch plugs) that block off the ends of some of the drillings.
Once you have a clean carburetor, the next step is to examine the gaskets and flat surfaces, to ensure there will be no leakage. Then assemble the carburetor correctly. Finally, install it correctly so that the gaskets do not leak air into the intake passage.
It seems relatively unlikely that anything not covered by the above would be causing the fairly severely lean mixture situation you have. Leaking intake gaskets are less likely than a blocked jet or drilling, because your carburetor works better at idle than at high speed. Simple air leaks to the intake passage normally cause more deterioration in idle quality than high speed running. (This is because the leakage increases with intake passage vacuum, which is high at idle and almost non-existent at full throttle.)
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