One more thing I should mention: the homepage has never ever appeared properly, under any circumstances, when clicking “Visit Site” from the WP dashboard.
I have purged the whole Cloudflare cache again. The website homepage appeared properly once in Brave browser in private mode, as it usually does, and impropely with all other browsers, even in private mode.
It seems to me that the homepage appearing properly just once and only in incognito mode is a clue.