It's been a week since I've been able to work on this deeply troubling issue. I am revisiting it now, as I tried to do a small tweak to this post (where images that used to be next to each other have suddenly started appearing on top of each other) and found that this too made the whole post go blank. There seems to be no way to prevent this happening, aside from restoring my 11th May version of the post. I counted the images, and found there are 18. However, when I add the YouTube video and the two four-image galleries I have on the page, it comes to 21. One over the magic 20 mark. So I deduce that the issue seems to be related to media in general, not just images.
Anyway, I feel very nervous about disabling all my plugins. I will lose all my setting if I do that. Is there no alternative? I deleted all my deactivated plugins, but that didn't solve the problem.
PS I deleted the final image in the photo essay referred to towards the top of this thread so that at least the rest of the photo essay could display, rather than having a thoroughly unprofessional blank post sitting there!