First off: what monitors are you using right now?
This is a multi-layered matter with many variables; I wouldn't categorically blame the monitors just yet. Harmonic distortion can mean a lot of things. What are you expecting to hear from it? What settings are you using? Is it a "reel to reel" type tape saturation? Is it a preset that only distorts a certain frequency range? Only the very top end? The 2-4 kHz range? Only the low end?
The effects can be quite different. Harmonic distortion can add warmth in the higher mid range, additional body, top end sizzle ... it can also lop off a bit of the high end sharpness, or actually alter the stereo image or reduce the perceived loudness. You should be aware of what you're actually trying to listen to. Things are sometimes easy to "unhear" if one doesn't know what they're supposed to hear.
Have you ever had your hearing checked? Perhaps you could try out one of those frequency hearing tests that can be found online. Those of course aren't exact and by no means scientific or medicinal tests, but can give you an idea of your actual hearing range. Depending on your age, you might find out that you actually aren't really able to consciously hear anything above 12 or 14 kHz ... it's very normal. Perhaps you're expecting a way too drastic effect from a subtle 18 kHz upwards saturated shelf?
Perhaps there's an issue in your gain staging. Maybe you're not hitting the plugins hot enough, maybe the input level of the signal that's fed into the plugin is way too low for the plugin to add anything perceivable or interesting.
There are of course differences between monitors and I think it's one of the areas where you do get what you pay for - but on the other hand, it's still all just sound, if you know what I mean, and I'm not fully convinced that you would be able to suddenly hear this unsuspected dimension of sound if you had monitors that cost much more than your current ones. It could also end up being a disappointment because while the new monitors are indeed better, you don't really hear what makes them that because your hearing (or "listening behavior") is still the same. 😊