Why does my paper circuit only work when I press really hard?

Yeah, the adhesive is your problem. Regular copper tape uses a non-conductive glue, so tape-on-tape through the sticky layer is basially an insulator joint. It'll sometimes half-work under pressure because you're squishing the glue thin enough for the copper to touch through it, wich is exactly the flickering you're describing. Some tape brands sell a version with conductive adhesive, but the cheap craft stuff ussually isn't, and there's no way to tell by looking. So don't feel bad, your instinct was right.

The standard fix is to fold the tape back on itself at the joint. Lay the first trace down, then where the second piece needs to connect, stick the second piece down overlapping the first, then lift its loose end, fold it backwards over the top, and crease it so the sticky side glues to its own back and the copper face lands copper-face-down on the other trace's copper. Now metal is touching metal and the glue is stuck to glue. It looks a bit lumpy but it works every time.

Your corners are also a suspect, but not because splices are bad once you fix the adhesive thing. The cleaner move is to not cut at corners at all. Copper tape bends fine around a 90 degree turn if you do it slowly and kind of roll the fold with your thumb instead of creasing it sharp. A creased corner can crack the copper and you get an intermittent trace that drives you nuts, which matches your symptoms too. Smooth rounded folds keep it one continuous piece of metal. Also burnish everything down hard after you lay it. Run a fingernail or the back of a spoon along every trace. The tape sits up on tiny ridges of glue and paper texture, and pressing it flat makes a surprising difference for a low voltage coin cell setup where there's almost nothing to push through a bad joint.

For the touch switch itself, folding a flap so two copper faces slap togehter works, but you can also do the version where the flap has a copper tab on the back and the pad below has copper facing up, so closing the flap puts metal on metal with no glue in the path. And if you've got a soldering iron out anyway, a tiny dab of solder on a splice melts straight through the adhesive and makes it bulletproof. I do that on joints I don't plan to take apart.

The brad trick saved me on mine. Two paper fasteners pushed thru the card, one holding a paperclip or a bit of card as the lever, and you get a proper pivot switch with real metal contact and no glue anywhere in the path. Also stopped fighting splices by just overlapping ends and putting a staple through both pieces, then flattening the staple legs on the back. Crude but it's been dead reliable on the two cards I built for my kid's school thing. The flap switch I made first kept failing the same way yours does untill I switched to metal-to-metal everywhere.

I tried the fold-over trick on mine last night and the flicker went away, so that part checks out. The lumpy bits bugged me though, so on one joint I just laid a small strip of tape over the top of the seam and pressed it down hard, metal side up, like a bandage. Worked fine so far.

Question though, for the beep module part. The little plastic card module has two flat metal tabs sticking out the bottom. Can I just stick tape over the tabs to hold them down against my traces, or do the tabs need to be clamped some other way? I don't want to melt anything on that board.

The brad trick is the way, agreed. I did the same thing for a flap switch on mine except I used brass fasteners instead of plain ones, since the galvanized steel ones tarnish after a while and get flaky again. Splayed the legs flat on the back and put a strip of packing tape over them to hold everything down. Still working months later with no cleaning. For the beep module tabs, a bit of double sided foam tape under the card presses the tabs against the copper pads and takes care of contact pressure without you holding anything.

Update: got it working. Went with the brass fastener switch, splayed the legs flat and covered the backs with a strip of tape so they stay put. Paperclip lever on top makes a clean contact now, no thumb needed. Also folded the tape ends over at each splice so metal meets metal, and the flicker is gone completly. Moving on to the synth prototype.