Excellent point John R about ignition condensers, the spark does not happen without one. Sure, it reduces pitting on the points, but its also used to make a resonant aka "tank" circuit with the coil. That is what sustains the arc during the so-called "firing line" part of the spark event. Current is rushing in and out of the coil and condenser, dumping that initial condenser energy into the spark.
Speaking of condenser energy, the 500mfd "horn" capacitor charged up to 6V holds 0.009 Joules of energy. A Joule is a Watt-second. So that 500mfd capacitor can supply one watt for 9 milliseconds. More likely its supplying 9 Watts for 1 millisecond. That ain't gonna run the horn very long or give much of a boost.
I note my 1965 parts book shows the "horn" capacitor on the ignition page. Also note that nothing changed in 1963 with the capacitor running change at bike serial # 3015. Same regular, same horn pack, same generator. I thought the factory might be "getting ready" for the 12V Bosch regulator used on 65 and later XLCH, but a better explanation is given by
a guy on the AMCA forum that says "The horn capacitor was added to the XLCH to fulfill Federal requirements that stated the horn must blow at any time which it would not do at slow RPMs due to the batteryless system. The capacitor held enough charge to allow it to honk, once... "
I can find no "Federal" requirement, the FMVSS (Federal motor vehicle safety standards) that require the horn to work at idle. There is an Ohio requirement, "Every motor vehicle shall be equipped with a horn which is in good working order and capable of emitting sound audible, under normal conditions, for a distance of not less than two hundred feet." So maybe "in good working order" was interpreted to be "at idle" by some lawyer. The fact that the capacitor was a running change does imply something
made Harley do it, it was not something they wanted to do.
In the tradition of "a little right, a little wrong" on the internet, the capacitor cannot hold near enough charge to get the horn "to honk." Like I said, hook it to 6V battery, than remove it and touch the wire to the horn and I would be very surprised if the horn did anything. As usual, Dr Dick is on the right track. What the capacitor does is smooth over the regulated armature voltage. There are two periodic things going on-- the "bumps" in voltage that come off the generator commutator, and those vary with engine speed. Then the horn is a periodic thing, with its frequency dependent on applied voltage. So at low RPM its likely the two things interact, that the bumps in the generator output interfere with the operation of the horn.
The capacitor smooths out the generator voltage bumps and probably makes the regulator work better. I have tried for 25 years to make a Harley voltage regulator that works with no battery and no load and no capacitor. I gave up. Its really ugly. When the relay (or a transistor like I was using) open to turn off the generator field, the energy in the field has nowhere to go, no battery, no lights no load at all. So with a "catch" diode it dumps into the armature, making a 120V spike. With no diode, like a old mechanical regular, its only the field coil that snaps up to 120V or so. I have dozens of 12V generators, and I think I can run the 6V generator on my 1952 K on my test bench to see exactly what is going on.
Another AMCA forum thread maintains the capacitor was used to keep the lights from flickering at idle. That might be true as well, but once again, I can't see 1 watt for 9msec doing much to power the headlight, it probably just stabilizes the voltage between commutator segments, just enough so you can't notice the flicker.
As to the problems of the OP, yes, this capacitor will help your horn problems. Electrolytic capacitors like this one are the worst reliability components in electrical engineering. As they age they dry out and stop holding charge. So please drop 25 bucks for the V-Twin replacement one to see if it helps.
I will get my generator tester set back up now that I am getting settled in Florida. I will look at the effect of this capacitor, and I can at least look at re-pop horn packs to see how they act.