Why does my copper tape tear apart when I weed between the traces?

It's mostly the corners and the cutting depth, not the tack of the tape. The adhesive between your traces is continuous, so if the cut lines are clean and separated, the waste copper should lift as one sheet even with a strong tack. When it shreds into bits instead, that ussually means the cuts aren't fully penetrating in places, or the blade is dragging sideways and tearing the copper internally along the path. A cut that's almost thru leaves little webs of copper that snap instead of peeling cleanly. I run my blade just barely into the carrier film, deep enough that I can see tiny puncture marks on the backing when I hold it up to a light. If you're kissing the backing in spots, those spots glue down and you get exactly the tearing you're describing, so a slightly deeper cut is usually safer than a shallower one here. The tradeoff is that too deep and you start cutting the carrier, which makes weeding fiddly in a diferent way.

For the corners, yes, you can round them without redrawing everything. Most cutting software has a corner smoothing or fillet setting that adds a small radius to every sharp turn in the toolpath. Even a tiny radius makes a huge difference, because the drag blade physically can't pivot instantly, it swings wide on direction changes, and a razor corner forces it to gouge sideways right where your trace is thinnest. If your software doesn't do it, adding fillets to the traces themselves in your CAD or PCB tool works too and honestly looks better at these sizes anyway. One more weeding trick that helped me alot: score the waste copper into a few big sections with extra cut lines before peeling, rather than trying to lift one giant connected sheet. Smaller pieces are less likely to grab a trace on the way up. And pull slowly at a low angle, almost parallel to the board, never straight up.

I tried something similar with craft vinyl once and warming the sheet a little with a hair dryer made the leftover stuff come up way easier, the sticky stuff seemed to let go smoother. Not sure if heat would mess with the copper though. Also when I pick out the scraps I use the tip of a pin to lift one edge first instead of grabbing with fingers, that helped a lot with the tearing. When you pull the waste off, do you use your fingers or some kind of tool?

I'm in a similar boat but coming at it from the sticker side of things, I cut stencils with an old craft cutter and never thought to try copper tape for boards. Reading this makes me want to mess around with it. One thing I do with delicate cuts is put low tack transfer tape over the top before weeding, it holds the pieces I want to keep flat while the waste comes off. Would that work here or would the adhesive on the transfer tape fight with the copper's own stickiness? Also still figuring out what blade offset even does on these drag cutters, so if that matters at this trace width someone please tell me.

One thing that made a big difference for me was peel direction. I started weeding so the waste comes off parallel to the traces instead of across them, and the tearing pretty much stopped. When you pull across a thin trace the adhesive has to shear right at the narrowest point, but pulling along the length lets each trace release a little at a time. Also don't underestimate a dull blade, I swap mine way more often than I feel like I shold and the cut quality on copper drops off fast compared to vinyl. A cheap pack of spare blades solved that.

One thing I'd check if your traces come from laying tape down in strips rather than cutting a single sheet: the seams. Where one strip overlaps the next you've got doubled thickness and a different adhesive layer, and a cut that penetrates cleanly through single tape will barely score the overlap, so the waste tears right along the seam line and takes traces with it. Also worth verifying that your cut lines actually intersect at junctions, becasue a gap of a few thousandths between two pass ends leaves an uncut bridge of copper holding the waste togather, which shreds exactly like you describe. I zoom way in on the toolpath preview before cutting for that reason.