Most people have worked out how to change gears, and this helps a lot.
But seriously, Airheads have a short stroke, short rod engine and are not designed for low revs, and this is quite clear in the riders handbook, if you have one.
Any problem with tune will also show up much more at slow revs / small throttle openings and the only answer to that is meticulously checking everything.
I think the timing should start to advance before 2000 revs so you may have a problem somewhere , and the factory full advance setting is is also really only a starting point - the best spot could be 2/3 degrees either side.
After 30 years the needles and jets might be a bit worn and causing it to run a bit rich too.
Obscure faults that I have experienced have been a resistor plug caps with four times the resistance, excessive voltage drop through a well worn ignition switch, gummed up passageways in the carbs, a new plug which missed under load, and a partially clogged muffler.
Setting the fuel level in the bowls by measuring the depth also works better for me.
A stiff dose of Auto injection cleaner in the tank cleaned out the carbs, but some of the others took a bit more finding.
Sorry if the previous post caused any offence - I usually post last thing at night when I get back from the pub and as they say they don't always look as good in the morning!