> Big Band music made it a spectacle of keeping so many instruments in time and tune with each other.

"tight" big band music "in the pocket" is deliberately and consistently behind the beat.

Peggy Lee and other great singers are more in tune than auto-tune or any perfectly tuned instrument is. Modern tuning, aka equal temperament, is a compromise. Every key is perfectly identically subtly out of tune. A great singer will sing with natural tuning, adjusted on the fly for each key change. Peggy Lee would sing a third slightly flatter than an auto-tune will. Unless an instrument is also playing the third, in which case she would sing in unison with the instrument.

Most of us aren't Peggy Lee or the incredible studio musicians that came out of New Orleans. We need to learn the rules before we can break them.