Thanks for your response, Dan.
Yes! This is an issue with in Apples newer MacOS EFI/SMC updates.
When I posted the above, I thought it was a problem with my MacBook Pro; I've since learned that it's a common problem afflicting hundreds (at least) of Mac users. It is beyond ironic that the software Apple offers to its users to diagnose problems with their Apple hardware… doesn't work. So we're all waiting for Apple to get off their collective posterior and fix it.
There's a ton of people encountering this Unable to launch Apple Diagnostics Apple altered the EFI so all versions of MacOS are effected from Sierra on up is where I've seen it.
Yes, I also posted my query at Apple Discussions: MacBook Pro 2013 - cannot load Apple Hardware Test, and was referred to the Unable to launch Apple Diagnostics thread (which now has 163 "I have this problem too" clicks, while mine has 5). See my posts here and here for more detail and follow-ups.
I spent an hour and a half on the phone with an Apple Senior Advisor on Tuesday (May 28) about this issue, trying everything he could think of (he apparently hadn't heard about this before), ending with sending a bunch of information, which he said would be forwarded to some "engineers", who would get back to me in a few days. I provided links to the two Apple Community threads and emphasized that there are over 150 people now waiting for an answer. I'll post whatever I hear in the Unable to launch Apple Diagnostics thread, for anybody who's interested in following the issue.
The Apple Hardware Test (AHT) repair doens't fix this in Mojave.
Not sure what you mean about the "AHT repair"; the link is to a very helpful post at GitHub that tells how to get and use AHT for various old Macs dated prior to June 2013, when AHT was morphed into Apple Diagnostics, which is now available only over the Internet (or from the Recovery partition in High Sierra or Mojave, as I discovered in my research). See my second post in Unable to launch Apple Diagnostics for more detail about AHT/AD. Anyway, AHT doesn't "repair" anything; it's a simple diagnostic utility – and it won't load/run in any of the Macs that Cannot load 'EFI/Drivers/TestSupport.efi'.
Apple holds the cards for their diagnostic software. You could try calling them most likely they'll send you to a authorized service center if there's one near you.
Oh, yeah, about that… All this started for me in early 2018, when the intermittent crashes etc. I'd been experiencing with this Early 2013 15" MacBook Pro finally became intolerable. I didn't want to be without a computer while I was figuring it out, so I bought a 2015 MBP (still new with warranty, found on eBay). I then took the 2013 MBP to the local AASP, asked them to diagnose it, for which they charged me $60. Presumably they ran Apple's in-house diagnostic utility/ies.(The guy said something about the "EFI version"; I didn't know what that meant at the time.) They told me the 2013 MBP was fine. (The new 2015 MBP turned out to have problems also, and I ended up struggling with both – along with my long-term chronic illness – all year, until finally early this year I got together enough energy to try to deal with them.)
But my MacBook Pro wasn't fine. So I bit the bullet and did some research, learned that both Apple Hardware Test and Apple's pre-2015 in-house diagnostic software Apple Service Diagnostic could be found on the 'Net. So I got them, figured out how to use them, and they both found both several memory problems and a few others that are more of a mystery to me (I'll be posting those elsewhere here to ask if anybody can figure them out).
I was able to run both AHT and ASD on the 2013 MBP several times, until suddenly in the last month it told me it Cannot load 'EFI/Drivers/TestSupport.efi'. (It still runs the OS version of ASD, but won't run the EFI version.) It seems likely the fact that I'd installed both High Sierra and Mojave on it a month or two before, which may have changed the Boot ROM version (I don't know what it was before) might have caused this problem. Although, as noted in my post at Apple Discussions, I recently experimented by installing Mojave on a 2014 MBP, which didn't change the Boot ROM version, and didn't cause the Cannot load 'EFI/Drivers/TestSupport.efi' error. So we still don't know exactly what's going on.
Anyway, the nearest Apple Store is too far away for me as well, and my experience with the local AASP does not give me much confidence.
The ball is in Apple's court.