Solved
I went to the range again today after another thorough cleaning and ensured that the rings were not aligned. First 30 rounds went okay, but it would never lock open on empty. Then, it started jamming, then it started not picking up the next round at all (shorter and shorter stroke). It was exactly the same as my first experience.
So, I took out my friend's BCG that he let me borrow and noticed a difference right away - it was covered in grease and oil. I put it in and everything was different. The gun felt like it could breathe suddenly. It locked open on empty and functioned flawlessly.
So, I went back to my BCG, but first put gobs of oil all over it. Well, maybe not gobs, but I put significant droplets all over the BCG and the points where the bolt goes into the carrier and smeared it all around. Sure enough, it functioned perfectly after that.
I've owned dozens of guns, but never an AR. I have always cleaned my guns well and left just a light glaze of oil on the moving parts, never actually putting drops of oil onto it. This has never caused a problem and every gun I owned worked perfectly. But maybe AR's just have tighter fits, particularly when new, and need additional grease or lube. I own two Mini 14's and neither will jam under any circumstances, probably because they have loose fitting parts.
Well, I'm going to buy some grease and start using that on the BCG. Thanks for the input though, I learned a ton about AR's.