Yeah, he was a huge fan. That IS him in the crowd shots during Gimme Shelter. Some other evidence: He and his girlfriend Janis met Keith at a bar in New York (Nobody's ?), late 60's/early 70's and saved Keith's cig pack when he left. He participated in the Stones Tribute night at Max's in '78 that also featured Richard Lloyd, Robert Gordon, Lenny Kaye et. al. He also regularly covered Play With Fire and As Tears Go By in his acoustic sets and I'm sure there are other examples that don't spring to mind right now. Dolls drummer Jerry Nolan (also RIP) spoke of running into Keith in N.Y. not long after JT'S passing and receiving some conciliatory and heartfelt words from the man re Johnny.
I always thought it would have been great if the Dolls actually got to open for the Stones in the early 70's. We know Jagger went to check 'em out and was unimpressed but I think it would have been a great bill. Okay, now cue someone explaining that the Dolls were not fit to wipe the Stones shoes. While I wasn't old enough to see the original band in their prime (seen the Thunders-less reconstituted Dolls four times, all great), based on the recorded evidence they rocked and , more importantly, ROLLED on a good night, just like the Stones.