Why would a reamer grab the hole and wreck the surface like that?

The spiral tear and the part getting sucked up against the cutter tells me you're reaming with too much stock left. For a 3/8 reamer in mild steel you want the pilot hole only about 10 to 15 thou undersize. Most people drill like 23/64 or a 9mm for that and it cleans up fine. If you drilled something like a 5/16 pilot, the reamer is doing way more work than it's designed for, and straight flutes grab hard when they're overloaded. The flutes load up with chips, there's no room for them to escape, and the whole thing chatters and tears.

I fought the same battle doing dowel holes in some brackets last year. What ended up making the difference for me was a cheap spiral flute chucking reamer instead of the straight flute one I had. The straight flutes would dig in every time I hesitated on the feed. With the spiral ones it just kind of screws itself in smooth and the chips clear themselves. I feed it in one steady motion on the drill press, no dwell at the top, and pull it back out to clear chips partway thru. Night and day on finish. They're not expensive either, might be worth having one on the shelf before you give up on reaming entirely.

Before you blame yourself entirely, mic the reamer itself. I chased a similar problem once and it turned out the reamer was a couple thou undersize from new, so it was basicly burnishing the wall with zero clearance and that alone will make a hole grab and tear. Cheap ones seem to vary a lot. Also worth putting a small chamfer on the hole mouth after drilling, since a sharp edge tends to let the lead in crooked and then the flutes are cutting lopsided from the first turn.

One thing I learned the hard way on my drill press: don't ever stop feeding mid-cut. I'd ease off when it started squealing, and that hesitation is exactly when it'd bite in and tear. Just keep steady pressure and let it finish. Also, slow isn't always better. Reamers don't need drilling speed, but crawling along lets them rub instead of cut, and rubbing is what burns up the oil and glazes the hole. A mid-range rpm with a firm, constant feed worked way better for me than babying it.

Quick update — I decided to test the theory about the steady pressure. Instead of backing off when it started to whine, I kept the feed constant through the end of the cut. Surprisingly, that stopped the tearing and the material grabbing. I'll try that approach for the rest of the batch.