I think there were two reasons for Jon to bring up Sting's play:

For an excuse to bring up David's Tony Award and Memphis - which is a nice little gesture.

But I think the primary reason is that he felt moved by the passion that Sting had for his play - which encouraged Jon to see the beauty in having the 3 guys (4 including Shanks) formulating a song in the studio together - the way they used to. I think he's drawing a parallel between Sting's hard work and attachment to his play, and the band's renewed passion for the craft of writing and recording together. At least I hope that's what going on in the BJ machine!