Just as a follow-up (thanks Carl!)...

I continued the shocking and constant filtration, while pouring 6 cupfuls of ALUM in the filter basket after each backwash. The Taylor kit finally arrived (having been constructed on the moon, apparently), and it showed my alk to be much too low.

I know you said no to Alum in the pool, but as the evil pool store guy explained, if you dump it in the skimmer, it forms a sticky barrier on top of the sand. I can't say for sure if it actually made a difference, but the backwash water was always a lovely brown (while water in the pool was even cleared up).
Something was doing something.

So now it is normal.

(I should add that I had an analysis done by a different store before the kit arrived. They wanted to sell me the hardness increaser and I refused, based on your advice. The kid at the register was dumbfounded, lol)