The first thing that come to my mind would be most of the songs recorded by The Velvet Underground. Many were lo-fi, but had an "aura’ to them. They would have lost their charm IMO had they been done more “professionally.”
Lots and lots and lots to be posted (yeah, often the production and the sound make the record), but as an extreme example I propose the whole of My Bloody Valentines’ classic album “Loveless”. The studio is the most important instrument on this record. The production costs almost ruined the label, Creation, by the way…
ETA: a collection of songs defined similarly by their production is “Pet Sounds” by the Beach Boys.
I’d add “Layla” to this list. The entire album is pretty intentionally raw and personal. For most songs, Clapton used small amps set up on chairs. “Layla” itself has a lot of overdubs, but the actual presence of the artists is very palpable. It is great, but not slick.
Mazzy Star was my first thought. Since you took that, I’ll go with Cowboy Junkies “The Trinity Sessions.” Entire album recorded in the Church of the Holy Trinity on one mic.
“A Day in the Life.” That combination of vocals, instrumentation, lyrics, production—the whole vibe—depicts an existential dread that strumming the same song around a campfire (or different production) just could not.
As albums go, Low by David Bowie springs to mind. But it’s far from the only time a Bowie album had that sort of - uh - sonic integrity? Station to Station is another, certainly.
Two good examples. I don’t know how he did the trick, but “Station to Station” was a miracle album although Bowie much later admitted that he couldn’t remember anything about recording it because of the drug haze he had been in.
Another song comes to mind is Crying To The Sky by Be Bop Deluxe with Bill Nelson’s terrific Hendrix-influenced guitar work and those two breathaking solos.