How do I hang a heavy zigzag shelf without wrecking a friend's wall?

A French cleat is actually the right call here, but not just becasue it spreads the load. The real benefit is that the cleat is a seprate board, so you can make it longer than the shelf itself and hit two studs even where the shelf doesn't line up with them. Screw the wall cleat into the studs, screw the mating cleat to the shelf, and the weight hangs on solid lumber instead of drywall. For a 30 pound shelf with books on it, that's the difference between fine and eventual disaster, because books get loaded and unloaded and the real weight is never what you calculated.

One thing about finding studs, since you said you can't locate one. Cheap stud finders misread all the time on drywall, expecially if there's a texture skim coat. Try the knock test, or measure from a corner, outlets are usally mounted to a stud so measure 16 inches over from one and probe from there. Walls are almost never actually 16 on center from where you'd guess, so measure, don't assume.

If it turns out you truly can't get a stud where you need one, forget the plastic anchors entirely. Get snap toggles or the spring wing style toggle bolts. Two of those will hold way more than 30 pounds in plain drywall, and you can run screws thru a cleat into them so you still get the cleat advantage of spreading the pull-out force along a wider strip. On your half inch plywood question, it's fine, no weirdness about it. That's how plenty of floating shelves are hung. Just don't rely on the plywood edge grain, screws in ply edges pull out easy. Best move is to screw the cleat to the shelf from inside or from the back, going into the face of the ply, with a bunch of screws, or add a strip of solid wood inside the shelf carcass and screw into that. Half inch ply in face grain holds screws plenty strong for a hang-it-on-a-cleat situation where there's no outward pull, just downward shear.

Honestly once the cleat is on studs, hang the shelf, level it, done. It'll never move.

Been in the same spot with a loaded bookshelf and the cleat saved me. One thing I'd add, make the wall cleat a hair shorter than the shelf one so the shelf sits flat against the wall instead of rocking on the cleat edges. Learned that the annoying way and had to shim it back down.

Also check for the wobble after you hang it. If she's got kids or a cat that'll climb it, add a screw or two through the shelf back into the cleat once it's level, stops anyone from lifting it off accidently.

Following this because I hung a heavy mirror at my place last year and went through the same anchor confusion. Ended up using the toggles and honestly once they're in they feel rock solid, way more than I expected from drywall. One thing I did that maybe helped, I hung the empty shelf first and put my full weight on the front edge for a bit before ever loading it up. Then added stuff a few things at a time over a week. Kind of paranoid I know but it made me feel better before my wife started piling plants on it. Did you allready cut your cleats or are you still in the planning stage?

Update: went with the cleat and made the wall side longer like the first reply said, actually found one stud within reach so I'm hoping to catch a second one with the extended board. Still stuck on the other end though. Do I trust a toggle out there or leave that side hanging off just the studs? Shelf sits about four inches past my last screw.