I've messaged a few people who show a close DNA match with no obvious matches in mine and/or their trees but had no reply or a dead end from most.
This might be the area you need to explore further. If science says there is a high probability of a common ancestor then working out who that could be is a priority. Perhaps the other people have been working on the basis their facts are correct without considering alternative options?
If the father was involved in a relationship with another woman around the same time and had children with her, then it isn't unreasonable to consider the possibility there might have been other relationships that led to other children who perhaps always believed someone else was their father.
It will probably be a case of doing more work on the DNA matches and looking carefully at the localities where the different people lived to see if there is any commonality.
I do think my grandmother might not have been sincere when she said what she said in the graveyard, although we go off of it because we've nothing else to go off.
This is where it gets really difficult. It is possible that what your grandmother believed - presumably what her mother told her - wasn't the full story. So she might have sincerely believed it, but that doesn't necessarily make it correct.
What is interesting is that if she
really didn't want anyone finding out then why say anything at all? It could either be that she
did want people to know and investigate (so her father wasn't forgotten completely), or it could be that the information was intentionally misleading. That's why I asked the questions in my previous post - understanding the "
why" part might lead to other clues.
The other part is the motivation for the secrecy. It is easy to assume that embarrassment about illegitimacy was the reason, but it might be worth thinking about other reasons why someone might not want details of their father to be known, but possibly still not want them to be completely forgotten. Relationships and emotions are very complex things.
You said Annie Elizabeth went on to marry someone else. Have you explored this man's history? Have you traced his tree to find ancestors, siblings and descendants? Did they all live in the same area?