Is compressed insulation in a 2x4 wall a dumb idea?

I did basically the same thing a few years back, stuffed compressed R-19s into 2x4 walls during a remodel, and it worked out fine. Your understanding is right on the compression part. Squished fiberglass doesn't lose its insulation ability, it just performs at whatever thickness you actualy end up with, so R-19 flattened into a 3.5 inch cavity lands somewhere around R-13, maybe a touch better because compressing it adds density. Since it's free, that beats paying for R-13s every day of the week.

The real risk isn't the compression, it's the install. A batt that's too thick for the cavity wants to bow and puff back out, and if you just shove it in and slap drywall over it, you get waves of it pressing hard against the sheathing in spots and hanging loose in others. That's where you lose performance, not from the compression itself. Take the time to cut each batt to fit the bay cleanly, split it around outlets and boxes, and fluff it so it fills edge to edge without being jammed. Fisher cuts around boxes help a lot. If you're careful, it'll do exactly what you want.

One thing to check before you commit: is his leftover pile enough to cover the whole wall, plus extra for offcuts and the pieces you'll waste around windows? Nothing worse than getting 80 percent done and realizing you're short and buying full price bags anyway. On the facing, yes, paper toward the living space in a heating climate. The vapor retarder belongs on the warm-in-winter side. And if any of those batts have been sitting in a damp garage or got rained on, feel them first. Fiberglass that's been wet and dried repeatedly can clump up and never fluff back right. Last thing that matters more than the R-value either way: seal the air leaks before the drywall goes up. Foam or caulk the gaps around the top and bottom plates, around wiring penetrations, and anywhere the framing meets the exterior. Cold moving air through a wall will undo good insulation fast, and you'll never have this part of the wall open again.

One thing that bit me with oversized batts is the width, not just the thickness. Mine were sized for a wider stud spacing, so every single one needed trimming down the side. Cutting fiberglass batts clean is its own skill. What finally worked for me was compressing the edge with a straight 2x4 and slicing along it with a utility knife, running the blade tight against the board. Free hand it turns into a ragged mess and you waste half the pile. Also staple the paper flange to the face of the stud every foot or so instead of shoving it in unfaced, keeps the whole thing from sagging behind the drywall.

Check wether those leftovers are even faced, garage extras often are unfaced and then the paper question is moot. When I did my bedroom redo I grabbed a cheap hammer tacker and it made the facing side painless, just staple the flanges every few inches as you go so nothing sags before the drywall lands. I stapled to the stud faces instead of trying to inset the flanges, faster and it kept the batt snug against the back of the cavity. Only annoyance was staple edges catching the sheet later, so I made sure to sink them flush. Either way with free batts you're way ahead of the game.