UPDATE: I filed a set of cups just like in John's picture below. Worked wonderfully! I oiled the threads before threading in, and all the gunk accumulated in the flutes. I repeated the process and after the second time, no more gunk!

I think what happened is that the media hadn't been adequately blown out from the voids in the frame before powder coating. Somehow, even after facing, bits of media had remained embedded in the threads. Or, when the facing got done by the frame builder, the threads of the tool didn't go past the first 10 or so threads. the SKF BB unit I installed has very deep cups that thread very far into the shell, 15-20 threads, farther than typical Shimano cartridges. After the chasing, it threaded in by hand very smoothly!

Problem solved. Thanks to all who contributed.

For what it's worth, this frame is a new frame from a very reputable and long-time local frame builder. He knows what he's doing. It's not my frame, but a friend's frame that I am building up.