Deactivating the Jetpack plugin completely fixes the problem with Tailor.
I re-activated Jetpack, then I deactivated all the individual plugins, and the problem was still fixed.
I then re-activated things one at a time. I stopped when Tailor stopped working.
In my case, it was the Sharing plugin within Jetpack. With that disabled, Tailor works fine, but if it is enabled, the pages all become uneditable.
I can’t say if there aren’t other plugins within Jetpack that have similar effects, but I can say for sure in my case, Sharing was the one causing the problem.