"Is there ANYTHING better than live recordings"
That's a pretty open-ended question. I'm hoping to pole vault on my 90th birthday. That would be pretty hard to beat.
Related to the music, IMHO it depends on how important your engagement is with the performers vs the music itself as a "stand-a-lone" entity. If you welcome a high degree of emotional interaction with the musicians (personality, intimacy, energy, great hair) your enjoyment may be enhanced by those factors. It you tend to prioritze getting the music "right" then a meticulous rendering of the the performance may tip the scales in your favor. That being said, if you combine those 2 factors, you may forgive a less than stellar performance/recording if the heart of the performers shines through. So, doing the math, a performance/recording of a "4" combined with a stage presence of "9" may tip the scales in the favor of that recording vs a solid "8" studio recording. A "10" plus "10" live performance would take extraterrestrial interfernce to produce a superior studio recording.
When a visual is introduced, the whole process gets more complicated. A "3" performance + a "9" energy/presence + a "9.5" visual can produce a performance that can result in the most fun you can legally have at home.