That's why usually good audio tests use ABX where it's not about seeing if you can identify what is best, but rather if you can identify a difference at all. See for instance: http://abx.digitalfeed.net/
EDIT: I actually just tried the test and... ended up doing the same thing you did. On the Neil Young track I could hear a definite difference in the high pitched violin-like background, but I ended up selecting the 128kbps track as the "highest quality", maybe because the compression made it sound smoother.
If you can't hear the difference at a low price point, that's not a curse, it's a blessing.
The DAC chip in a monitor will be the absolute cheapest one they can source, and is probably receiving a lot of noise from the monitor itself as well. The good news is that you don't need to break the bank to get something decent; a budget of ~$100 will get you something a million times better than a screen could ever deliver.
The $400 Element II [1] from JDS Labs (makers of the legendary Objective-2, a DAC+Amp that was designed to give top-end performance at bottom-end prices) is very literally as good as it gets without snakeoil entering the equation. THD when under a 150-ohm load is 0.0008%. That's very good. This is what I'll be upgrading to if my setup at work ever dies.
My work rig is an Aune T1 [2] which retails for $250 but Massdrop does them for $100-$150 sometimes. Yeah it's a "tube" but that doesn't matter as much as myth would have you; today's tubes are pretty transparent unless you deliberately get something that isn't. THD is good however nowhere near JDS Labs-level of perfection.
The Fiio Q2 mk2 [3] can be found for as little as $100 and it's a solid performer, though I haven't personally used it, Fiio's reputation is impeccable.
[1] https://jdslabs.com/product/element-ii/
[2] https://www.amazon.com/Aune-Second-Generation-Amplifier-Deco...
A man sits on a weight bench, wondering if the equipment is defective because he can't seem to bench press 800 lb...
It's a big step between esoterics and "please don't use the cheap shitty DAC that was integrated into your monitor to satisfy a feature checklist".
The amp part will matter a lot the more impedance the cans have, and many of the better headphones are high(er)-impedance. Might as well spring for a headphone amp + DAC combo just to see how it sounds. It's not that expensive, FiiO has some for under 80 bucks, for example.
https://gizmodo.com/is-there-anyone-stupid-enough-to-believe...
[Edit: Fixed Link]
Yeah the AMP and the DAC.
The Apple USB-C to 3.5mm doesn't break the bank, works with Windows fine and the headphone nerds seem to rate it