How can a system be judged with highly processed, non acoustic music?

If you assume the "standard" is real, live music, it is harder to evaluate something that has been heavily processed or involves electronic instruments. But those can inform too.
From my perspective, anything done in a studio is likely to have undergone some "adjustments" whether they actually improve the sonics or not. (I’m reminded of the line that a lot of gimmickry goes into making something that sounds "natural").
Leaving that aside, if I listen to something like Crimson, Live Toronto 2016, I know what the actual band sounds like rendering that performance in a large concert hall, so the bass, for example, from Tony Levin, on the "stick" can tell me how deep and loud a system can go- whether that is of importance is a different question.
I listen mainly to small combo jazz, combined with early proto-heavy rock (kind of an odd juxtaposition, but there it is) and can only evaluate a system meaningfully by listening to a range of recordings. Having a "natural acoustic" that is being reproduced in your room, imposing the sound of the recording venue as part of the delivery, is only one of a number of aspects I listen for. I do like live recordings for this reason. I’m not listening for "thrills" but mainly for tonality and realism, but that’s pretty subjective.

Call me crazy. I tend to evaluate the sound of a system as to whether it sounds good with the music I like, from acoustical to electronica. Is there really more to it than that?

Obviously, recording music is an electronic process-mixed and mastered. However, many recordings clearly strive to recreate the real, original sound. I would point to “Audiophile Recordings” which, mostly, are acoustic and “natural”. Surely, such recordings are sweetened and adjusted but the goal is a natural SQ.

A room a studio, a concert hall even outdoors they all contribute to the sound you finally hear.When I was a kid on Saturday nights there was a bar across the street from where I lived  .The guys would sing Doo woop under the no 7 IRT train line the el was concert and it had an echo you could hear for blocks.These guys ,some of them anyway sounded so great.Just them sing under the El.......

Distortion, or lack there of.