Got the Digi-key order. Good news: Tip activator seemed fine, as did the flux and solder. I replaced the switch in the Bad Monkey and it works as it should now!
Bad news: I also replaced the 4053 in the Nux Horseman and that didn't change anything. It's possible I didn't connect all the SMD pins so I need to check that, but I also noticed that with everything dimed there's a very faint signal coming through. If I boost the signal 24dB I can hear that the gain and treble for sure have an effect on the signal as they should (reducing gain or treble when I turn them down). So I suspect the signal pathway is working, but maybe some gain stage is broken or something along those lines?
Fixing it gets a bit trickier now since I'm not sure if I can use an audio probe since it's stacked PCBs, I might need to find a cable that works for connecting the PCBs (like in the video) and if I figure out where the issue is, that I can then find out what that part is since they're all teeny tiny SMD parts.