I've fixed it!
TL;DR: We were planning on moving the Web Adapter to a new server anyway, so I did this and the trouble seems to have gone (fingers crossed).
I became suspicious that the problem was isolated to a single Web Adapter-ArcGIS Server pair, because switching the service to another pair immediately fixed the problem. I did some testing from an external (overseas) network and, although I got inconsistent results, it appeared that the connection to the original was being dropped a lot of the time. Generally, the following were true:
- I could visit /arcgis/rest/services with no trouble.
- Visiting any folder or service, e.g. /arcgis/rest/services/Waikato_Data_Portal/Waikato_Data_Portal/MapService/58, would result in the connection being dropped.
ArcGIS Server was responding as expected when connecting to it directly, so that put my focus on the Web Adapter. I'd been planning on moving the Web Adapter to a new web server anyway, so I took the opportunity and did this.
There was also an issue on the old web server where the intermediate SSL certificate was not installed correctly (see Sydney Bohn's comment and my response). I don't think this was the main problem, because I was having trouble on both HTTP and HTTPS but, on the other hand, SSL is tricky. In any case, it's no longer an issue on the new server.
So, I never really found the root cause, but a new Web Adapter fixed it for me. If we hadn't been moving web servers anyway, I would have tried reinstalling the Web Adapter on the original web server.
Thanks Sydney Bohn and Hemant Angarkar for your help!