Your pilot circuit controls idle to about 1/4 throttle. These carbs work on throttle position, not engine speed. If the bike is not starting on it's own then the pilot circuit, which includes the enricher/choke, is clogged.
Also, if the rubber plug is "melting" then you have some bad fuel issues. E-10 fuel is terrible on rubber parts. Letting fuel sit in a bowl will cause it to varnish, then (because E-10 is also a cleaner) once the fresher fuel comes into contact with it, it will cause the varnish to flake off and clog shit up. A piece the size of a tiny pin head can render the carbs inoperable.
Those carbs need to be stripped down and cleaned 100%. Not just spraying some cleaner across the throats, but freaking cleaned!!!