What I'm saying, then, is don't fear all of those colourations. Even with a subjectively nice clean sound, there will be some present.
Agreed. The distortion (not coloration) that I prefer to remove in some recordings is that gritty, dirty sound that you don't hear on the Rain On Me track. The main place where we view things differently is in how/where colouration and distortion overlap. I've been trying to think of a good mechanical simile, but I've failed to think of anything in the mechanical domain that's really a convincing analogue of what I see in the electrical domain. Gas springs and therefore speakers in small cabinets exhibit the effect long before they 'bottom' and cause that coarseness, but it's hard to explain, even then, the effect. "column of air" instruments exhibit it.
You mention FFTs, so you'll understand well spectral data. Even-order harmonics tend to sound like part of the intended sound ... complementary; sawtooth-like waves. Odd harmonics are those that sound harsh; square-like waves, and the ones it's clear you definitely don't want. Worse still by far are intermodulation products.
I don't know if
+++ this chart +++ this chart helps explain why odd-order, notably the higher ones, (clipping) harmonics can sound so bad. Even heavily distorted guitars try to avoid the worst of the upper harmonics. A
small amount of some can help.
There's a good Sound-On-Sound article
+++ here +++ that may or may not help you in getting the sound you want and may better explain than me what I've been trying to say.