I hope you are able to go ahead and have your DNA tested, tomjackliam.
There's been a major twist in the story I relayed in this thread, which reinforces Cas's comment about DNA.
Ever since my brother and I did our DNA tests (through Ancestry), I'd been baffled by a very large number of distant matches, most of which were clustered in one American state (I'm Canadian but the grandmother in question was born in the US).
Well, I don't know what finally clicked, but I decided to tackle the closest of those mystery matches again and eventually it all fell into place, and I think I've identified my grandmother's biological father (and it doesn't seem to have been her mother's husband).
I've reached out to this closest match (a grandson of my possible great-grandfather) and, guess what, he hasn't responded. I messaged him when I was just re-starting the search and therefore there wasn't anything shocking in my message. To be fair, I don't think he's seen my messages: one was via Ancestry (he apparently hasn't logged in for over a year) and one was via Facebook private messaging (and I don't know if that message has been caught in a security filter). Sigh.
My possible great-grandfather died more than 50 years ago, but he had two children who were born about 20 years after my grandmother was born, and those two children seem to be still living. So, if that DNA match of mine ever does respond to my messages, I'll have to tread lightly.
In the meantime, I've sent off some requests for official documentation that might bolster my case. I would want these anyway, but they might also help if I want to convince said DNA match, because it's a long and convoluted story of how this might have happened.