Good point about Jagger’s live singing. And the band isn’t very tight, usually.
Also to your point, try singing along with Wild Horses. In the opening line, “easy” is an F# above high C. That’s up there. It’s also amazingly delicate throughout. Maybe my favorite Stones song, along with Brown Sugar.
That's the reason they sounded like a garage band and proud of it. While Keith plays on the down beat as does the bass player, Charlie would play just behind and Mick sang just ahead of it. Listen to Charlie's crash symbols - they are micro-seconds later, along with his snare. This is what makes them unique in rock while most bands had all on same down beat, some instruments would get loss in the mix, which the producer would turn those up. Of course all Stones kept the same tempo (speed) and didn't do that off-beat thing on all their songs.
I could have the above off beat reversed for Charlie and Mick, but don't think so since I've played my drums to Stones classics. Run Honky Tonk Women through your mind from Charlie's intro of snare, tom, floor bass (crack..tum,bump - then his regular hi-hat, snare, bass pattern) and listen to when Keith comes in. Mick's singing is obviously ahead of Charlie and Keith.
Gimme Shelter is another good example. Keith and the female vocalist do the haunting into and Charlie comes in with his snare chha, chha, chha, ccha...chacrack/crash behind Keith. It continues throughout the song.
Yes, Wild Horses is up there in pitch and one of the Stones best slow songs. Also had a connected melody interlude to the rest of the song.