I honestly wouldn't worry about the alcohol thing - I've used it for years. It's absolutely fine for electronic contacts, especially with something as tough as a WICO joystick.
Wico joysticks are extremely robust, so there could only be a few points where the problem occurs:
1. the leaf switches inside the stick base (they get dirty or bent so they don't make contact). Solution: clean 'em or bend 'em back.
2. the button selection switch. Clean it, replace it, or bypass it entirely.
3. The cable. Broken wire. Replace the whole cable.
4. The 9-hole connector at the end. After a while some of the copper sleeves in the holes open up a little too wide and might not make contact with the male DB9 pins on the Commodore computer. Solution - either replace the whole cable or snip off the original DB-9 female connector and attach a fresh one. If you've got nothing to lose then you might want to squeeze the outside of the plastic DB-9 female connector hard with a pair of pliers. This might bend some of the stretched sleeves into an oval shape so they make contact with the pins in the male connector.
If I were you, yes, I would replace the cable with the one from the WICO boss. Really, you can bypass the switch entirely and have both buttons work at the same time. I don't know why they even included a switch. The switch is SUPPOSED to activate either the top or the base button (it evens says this by the switch on most of mine), but I have a few that are wired weirdly/wrongly like yours is.
But how many systems have you tested this Wico joystick on? One, two? And it doesn't work on all of them?
I had a VIC-20 that I had sporadic joystick response with. I had a hard time figuring out why sometimes joysticks wouldn't work in certain directions. I eliminated the joystick as the problem, as the issue came up with many different brands of controller attached to the VIC-20. The VIC's joystick port looked fine, so I tried replacing the input/output chips (CIA) on the motherboard. That didn't help. I re-flowed the solder where the joystick socket attached to the motherboard. That didn't help.
I then finally examined the joystick connector in the VIC-20 with a magnifying glass and saw that some of the brass pins on the inboard side of the socket, where they bend 90 degrees toward the motherboard, had severed completely. Most of the time the broken ends came to rest touching each other and the joystick worked. The breaks were almost invisible when this happened. But sometimes the stress of connecting or whatever caused the breaks to open up. I just bypassed the breaks with some wire and everything worked again.
The above solutions are dependent on the fact you're sure it's the joystick that's the problem, and not the socket on the computer, or the input/output chip inside the computer. Obviously if other joysticks work fine on this machine, then it's the Wico that's the problem.