It happens because you have problems with wp-cron tasks or DNS configuration.
If you have your website hosted locally, often the domain name is just recorded in /etc/hosts file. This is normally okay for browser access (and this is why indexing is coming well but slow when your WP admin is open). But PHP can not read /etc/hosts by some reason and thus indexing procedure can not go internally.
I just made a short check for Google Search results' appearance and I found that they are trying to extend the excerpt by other (not "good") sentences even there is only one "good" sentence and it's short (your case #1).
So yes, I think it would be a great idea to add one or two other sentences near the result to give more context information to the user.
At the same time, I know that some developers still want to keep excerpts as short as possible to keep more space on the screen for other results. So I think it will be a configurable option in WPFTS Settings.
Also, I would like to add a configurable option to put a text like "...other 5 appearances" at the end of the excerpt. Thanks for the idea!
We also have some other things to play with Smart Excerpts (for example, lots of developers asked me to add page numbers in case an excerpt was taken from the paged document), so I guess it will add those features one-by-one.
Also, the documentation for the plugin requires a sufficient upgrade, and I definitely will include this explanation there.
We have over 1,000 pdf documents on our website, and the most important is our Journal, with 238 editions at present. As an example, a couple of issues of our Journal are here, but they are all similar in structure: https://rchs.org.uk/wp-content/uploads/2020/02/Journal-100-Nov-1975.pdf https://rchs.org.uk/wp-content/uploads/2020/02/Journal-001-Jan-1955.pdf
You will see that at the bottom of the second page of Journal 100 is a table of contents for the issue and, if the pdf is saved and then opened in Acrobat Reader, there is an equivalent set of bookmarks. It would help if a search term appearing in the article title (as listed in the table of contents) was given a higher weighting than one in the text of the document, but quite often the term will only occur in the text, and not in the title at all.
Ideally, we would like the weighting to be based upon articles, but weightings based upon the the whole Journal is acceptable. This is because the articles tend to be on unrelated topics within a Journal, so there is probably little difference between the number of instances of a specific search term within an article, and within its parent document.
I think it would help me if you could explain, in non-technical terms, how the four weightings operate with WPFTS. I've looked at the TFIDF article on Wikipedia, and understand the basics, but the majority of the article is too technical for me. Perhaps this information could also be added to the WPFTS documentation?
As a related issue, could WPFTS open the document listed in the search results at the first article page where the search term is found? I realise it might go to the article title instead, and if so that wouldn't really help much.