I found there is no continuity from the output on the switch pcb to the hole next to it that goes to the jack.
In bypass mode, you wouldn't have continuity, but if you don't have continuity regardless of which way you have the switch thrown, then the switch is your problem. A bad switch could also cause you LED to not light up.
As for continuity between the Volume pot and the output, you would expect to have continuity on on pin 2, since they are directly connected. Checking for continuity across your circuit is only going to tell you if you have a connection between two components. The audio probe is going to be much more useful than continuity in finding the problem (unless of course it's just the foot switch). As
@jwin615 said, it's pretty easy to melt the inner workings of a foot switch, and when you do, you can get some unpredictable results. I desoldered one, and I thought I was good, everything tested fine except I had continuity between pins 4 & 6. If I had reused that switch, I'm probably still be scratching my head.
When you solder the new switch, don't sit on the pins for too long, and don't consecutively heat pins in the same row (vertically as pictured above). I jump around. For example, I'll heat the top left, then the bottom right, then the center, and just keep moving around until I get them all. Also, I'm not sure how hot you keep your iron. I keep mine at 680F, but if you're up over 700F, you have even less time to sit on those pins before you end up with a problem.