Perfect harmonic tunings sound incredibly exotic and out of tune to Western ears - not least because we're so used to slightly-imperfect equal temperament. We have decades of exposure to it, so it sounds normal. A perfect perfect fifth sounds very strange, even a little disturbing, and thirds are almost dissonant.
I don't think anyone who has even the first idea about this has ever suggested that harmonic tunings are a goal, an ideal, or even a practical possibility. (See also, Pythagorean Comma.)
I don't think anybody is likely to call a perfect fifth or third those things either. You can easily check using the "Generate, Tone..." feature in Audacity. I recommend using a sawtooth wave, which contains prominent high harmonics that make it easy to hear subtle difference in tuning. The worst criticism I'd make of perfect just intonation is that it can sound boring, for the reason the GP post suggests.
Many of the auditory things that trained musicians describe as creepy or weird, seem unremarkable to me.
I am a big fan of Terry Riley's - Harp of New Albion exactly because its five-limit just intonation piano sounds so incredibly weird to my ears.
I've met critics that even think that "pure" chiptune sub-genres such as the ones that stay within the bounds of physical NES hardware "can't" survive long-term. I think they are wrong both because I've heard some extraordinary things done within those constraints in my lifetime and for some of the reasons that "minimalism" will always be a viable subgenre of "Classical" (no matter how boring some people find that, too). But I understand where that idea comes from of how such a unique set of hardware constraints gave us some of the "purest" waveforms possible and why such pure waveforms are generally an anomaly in music.
Fender Rhodes and Wurlitzer electric pianos, for example, which are staples of pop and jazz, usually use what's called Stretch Tuning, wherein notes an octave apart are just slightly higher than 2:1 frequency ratios.
Old polyphonic synths like the venerable Oberheim OB-Xa would have their voices go slightly out of tune as the instrument warmes up. It's subtle, but once you hear it you start to notice the color it adds
If you've ever tried building an additive synth, this probably seems obvious. If you play two tones with exact ratios together, the brain has a tendency to interpret them as a single tone with a different timbre. By making the ratios slightly off (don't have to be by much at all), or by making the ratios vary a little over time (eg. Subtle vibrato), the brain is able to pick out the "consonance" as a chord, and it sounds far more interesting.