For model builds before 2013 the diagnostic test uses a different method.
Two different support documents cover most aspects; there are reasons
why the AHT vs Diagnostics may fail to work in certain build year models.
And with specific systems, the Recovery partition (or lack thereof) matters.
• Using Apple Diagnostics - Apple Support
For computers that shipped with OS X 10.7 or earlier, the test is on the
included system disc media kit; so you'd start from there to run test.
• Using Apple Hardware Test - Apple Support
Before You Run AHT/Apple Diagnostics. By controlling both the software and
hardware distribution for its systems, Apple is able to optimize the two halves to work ...
http://computers.tutsplus.com/tutorials/finding-hardware-faults-exploring-aht-ap ple-diagnostics--mac-60456
There are a few ideas and methods in the articles above.
A reason for the issue you face, is the vintage of hardware & macOS they shipped with.
Good luck & happy computing! 🙂