So, I have an update on the situation although I still don’t have any idea why or what may have caused the issue. I am not sure if the “correctly default the DAC’s configuration” that @Torq mentioned may have anything to with this, or if it is just black magic, but here is what happened.
First, let me point out that the DAC/Amp in question is the Hifiman EF400. The reason I didn’t mention the model previously is because I prefer to investigate possible user errors before anyone reaching the conclusion that the device is faulty (which is something that can happen quite quickly in the case of HFM products
). If it had indeed turned out to be an equipment malfunction (which it possibly was, then I would have still mentioned the model, just waiting until it was confirmed.
The model is actually quite important here because the EF400 has two USB inputs, one USB-B and one USB-C, which do not have a way of selecting between them. I don’t know how the internals deal with this, nor what would happen if both were connected at once. I haven’t tried connecting both at once, in my mind I don’t think it would be a good idea but there is no huge warning sticker to not do so, so maybe it has some kind of internal protection?
Anyway, as a quick recap, I had previously tried 15 cables, 3 of which were USB-C and the rest USB-B. With the vast majority of them (10x USB-B and the 3x USB-C) the output of both the headphone and DAC was ridiculously low. With the remaining 2x USB-B cables, which so happened to be Topping cables that came with DACs, the output was more powerful but still not powerful enough, I could still max the HD6XX without it becomeing painful (not comfortable but certainly no deafening).
So, this weekend I got another bunch of USB cables, of both the USB-B and C type. I started going through a couple of the USB-B cables first and the output was the same as with the 2x Topping cables. I then tried a USB-C cable, same story, and then I tried the USB-C cable that I use for charging and data transfer for my phone.
With this USB-C cable, suddenly everything was louder and I found that I could no longer max out the HD6XX, bringing my usual listening levels to around 9 o’clock on the dial rather than 1 o’clock that I had been experiencing.
So I had found a cable that worked. I then moved onto another cable, same result, everything was fine and at levels I expected. I then tried another USB-B cable and again, everything worked fine.
I decided to switch back to the ones that had been giving me issues and for some magic reason, everything sounded fine with them also. None of the cables that had been giving me lower outputs continued to do so 
As I had been doing all these tests with my laptop, from Foobar, I decided to go back to other sources and with all of them (PC’s, Android’s, Pi4) the output is now fine, with all the cables, including the ones I had issues with before.
As I said, I have absolutely no idea what happened and the only thing I can think of is that it could be realted to what @Torq was metioning, that the DAC had somehow got “stuck” in a state that was limiting the output, and that the cable from my phone somehow got it out of that state. I don’t even know if this is possible, I just can’t think of any other possibilities and I have been trying for the past 4 or 5 hours to recreate the issue I had previously without managing to do so.
I am very happy that this is resolved and I can now get on with actually reviewing the EF400 as I really like the way it sounds and now it performs as it should.
Thanks to you all for your input!!