I agree with OP, a live act should have a commitment to entertaining people. To me that is the difference between a good and bad live band. So the best live show I ever saw was Fela, and not because the band was tight, though it was. He brought a corps of African dancers, a full horn section, and the best performance I have ever seen. Worst was REM when they were starting out (I didn't go see them after that) and Young the Giant when they were starting out (they got so much better later, we saw them again and it was such a good show). Because they just sort of stood there. Hayes Carll and Paul Thorn are great to see live because they tell stories and are really funny. J Roddy Walston won our hearts the first time we saw them, ginger madman at the piano.

Recorded music is art but live music is performance art and entertainment. I don't think it's reasonable to say that because a band sounds great live, that makes it a good performance. That's a baseline (ha) not a goal.

Tool, the singer is known for being actively hostile to performing. A good sound and a dull show does still mean it's a sucky performance, I think.

I don't like performing either, but when my belly dance teacher dragged me along with her to paid gigs (because I could dance) you better believe I engaged with the audience, that IS the job, and that audience generally didn't even pay to be there. She said stage fright was selfishness - that your job was to give a performance, that it was about giving something to the audience, not being perfect - she would say it's not about us it's about them. If that is true dancing for a party, it's a hundred times more true for an act people paid a hundred dollars to see.