Trying to open the html manual directly from the website, I no longer get to the
cover and context menu,
...
This has been happening for a couple of weeks.
This is fixed so IgorBeg's workaround is no longer needed. It was my mistake for removing "index.html" so as to not conflict with "Index.html" (lower and uppercase "i" is the only difference). Although lowercase "index.html" is not used in Context Help, it is needed for the HTML manual itself on the website. I will see if I can find a way to change uppercase "Index.html" to another name for a different solution.
P.S. it is probably better to report a potential error right away -- preferable even if it turns out not to be a real error. I should have verified everything was working after I removed lowercase index.html from the website (shame on me!)