I have to tell that you that even downgrading to OS as old as Lion won't rectify this issue and likely nothing will ever. I never upgraded by following the common approach (installing on top), I always did a clean install on a separate partition: that way I independently run Lion, Mavericks and High Sierra to this day, the last two on an external SSD drive. None of them is able to boot into testsupport.efi. My Mac, in case you've not figured out, is mid2012 MacBookPro. I don't remember when was the last time I ran AHT from Lion which I used exclusively till the end of Dec 2018: it could be 2017 when AHT over the air was accessible. I have to add I've reinstalled Lion about 6 times in the course of the last 6 years I own the machine with the last time being Feb 2018. The strange thing is that just today I navigated to /System/Library/CoreServices in Lion and found .diagnostics folder missing while adhering to the common logic it should've been there. Couldn't it be that reinstalling even such an old system as Lion could modify CoreServices directory if Apple changed something in the recovery version for several Mac models, older than, say MBP 10,1, so that even reinstalling the oldest supported system would strip it of AHT? That's why I think that the last time I ran AHT was 2017.
Of course, I did all the steps mentioned in that post on GitHub: found the compatible AHT disk image, installed that hidden folder to the proper destination, set correct permissions - no result.
It would be interesting if owners of newer machines (2013 and up) chimed in and told us whether Apple Diagnostics is available to them.