ok firsts off you cannot rule out a battery by testing when the motor is running, my suggestion is, pull battery and have lead tested at a auto parts store, , its free and fast!
next odds are the wire on bottom of carb was for a carb heater, not many models have them but some did!
as for running better in reserve over basic run position on the peacock, YES that can make sense, as them peacocks can get built up with gunk just like a carb can, and cause flow issues
as can fuel lines over time, building up crab and causing poor flow to carb
and again, cleaning, rebuilding the carb and having dirty fuel, crab in tank, peacock and lines, can be a wasted time on clrb cleaning as again, crap flows down stream, making cleab carb dirty again!
I would again suggest cleaning tank/peacock, and I would even replace fuel lines there cheap and simple, and then TRY running, if still acting up, maybe pull carb again and clean,(or just do so all at once and know its all CLEAN and correct working order!) and again, new spark plug rules out a bad plug too here!
IF after doing all this> Oddd are its maybe the CDI, and NO test is there for them< but you need to rule out other things before going CDI route IMO