I've had to take my PS2 controllers apart to clean them when I was like 11. Have some faith in yourself. At least they don't have the spring-triggers like the following generations, unless your aftermarket ones do.
Also, this soldering job would not be hard at all. It can be done with a $1 flea market iron. Could probably re-use the old solder if you're careful about it.
That's not the Anarchy symbol. That's the button disable "Auto Start-Stop" in Ford & Dodge vehicles