The net-net on this for me is this:
1. I did everything than was required, multiple times, and it didn't work.
2. The sitemap_index.xml is supposed to be in the root directory - the link, created by the Yoast plug-in - that displays the file is looking for it in the root directory, so obvivously it thinks it was created there. It was not.
3. There are no permissions problems - the other sitemap plug-in I ended up using - Google XML Sitemaps - had no problem creating their sitemap.xml file in the root directory. I also successfully linked to it from google and bing without a problem.
4. I am doubtful that there is any conflict with another plug-in, since this is a very straightforward wordpress site. We only used two other simple plug-ins - Contact Form 7 and NextGEN Gallery - and neither should conflict in any way.
So, it looks like the Yoast SEO plug-in has a problem. As I said, I am still using it - it is the best SEO plug-in - and wish I didn't have to use another plug-in for the sitemap, but it is just not working for me, and it sounds like for others.
I would love to hear from Yoast with a fix for whatever the problem is.