Thanks for taking a look. My understanding was that the 'simulate overprinting' setting was the correct way to view these files, but my printer is insisting otherwise. Here's their message about it:
"I wanted to do was put into writing exactly what we are seeing from our side. The latest artwork that we are working with, has overprint on it. I’ve attached a few screenshots for you to show this, the White Overprint – No Black screenshot, is what happens when we knock the black out on the artwork we have, leaving the white print left. Sadly, no matter what we do, this is what we are left with when we try and send your artwork through for a proof, and we cannot remove this overprint. The Removed Overprint – No Black screenshot shows what happens when we toggle the option to remove the overprint, this is for show only. This is how we need the artwork to look when it is sent through to us, for reference, what you can see in this screenshot is a toggle only, and it is not a removal, as we do not have the function to remove this completely from your artwork. The final screenshot First File – Overprint Options Shown, is a full screenshot of this main page that has caused an issue, and in the box on the left, is say Overprint: Yes."
So is the problem with the printers and the PDFs are fine?
Hi @stuartm1980, I’m also not seeing any overprint problems with either your InDesign Sample exported to PDF/X-1a or the PDF/X-4 sample. Rather than using the Simulate Overprinting checkbox you can use Preview>Object Inspector to check the actual overprint property of the texts when you click on them:

Your ID sample exported to PDF/X-1a:

Also, it sounds like there is a communication problem—it seems like for some reason the printer wants to remove the overprint, which is clearly set in your PDF.
...when we try and send your artwork through for a proof, and we cannot remove this overprint....