It is around 2 1/2 years since I posted this question, and I thought I'd update with my experience since.
To test if the tire liners were the only problem, when I removed them I went with new Kendas, rather than higher priced tires, as I was thinking of doing at the time. The tested Kendas are K40-HP ( 26 X 1 3/8, or 37-590 ), with a recommended maximum pressure of 90 pounds, which I use at 65 pounds. Kenda has others tires of the same size (for instance plain K40), with a recommended 55 pounds max, which is unhelpfully low for riding on pavement. Two of the tires that had the problems before were the 55 pounders, at 55 pounds. The last one, on the back, was a 90 pounder, at 65.
I have had no tire blow outs, and no flats, since then.
Last summer, I looked over my tires, as I occasionally do, and saw a cut on one tire, and decided to change it. I used the opportunity to examined the tire with 2 years of use, and inside it looked identical to the new tire. No sign of gray and fraying areas that I got when using tire liners.
For frequent flats from glass or thorns, tire liners could be useful. As insurance against occasional flats, to avoid having to change tubes in the cold a long way from home, they are a new problem, not a solution.