From what I remember of his 1970s/80s biography, Sullivan publicly sparred exhibitions, carting Paddy Ryan (among others) around with him for that purpose. He must have also picked up something from the various fighters who traveled around with him for his grand show tours.
Going from memory, opinions, articles, and books I have read, with contemporaries of Sullivan quoted: John L. was in a class that nobody had ever seen before. In the early 20th century it was believed often by the fighters, themselves) that, in his prime, Sullivan would have bested Jackson, Corbett, Fitzsimmons, Hart, Burns, and even Jeffries. It MAY very well be that nobody better than Sullivan came along until Jack Johnson, or (maybe) Sam Langford. On the other side, would Tunney have even beaten Dempsey, with Jack in his prime? Most didn't think so at the time. Would he even have been able to beat Schmeling? Personally, I doubt it. Does that mean that Sullivan would have beaten Tunney? Maybe so, maybe not. But it's all worth considering, at the very least.
Sullivan himself thought that Jeffries was clearly a better fighter than him for what it's worth. Maybe he was simply kind for a young champion, but I think that Jeffries should be at least considered.
Jeffrries was one tough customer. No doubt about that. As for Tunney, from what I've read, Sullivan was a cross between a Swarmer and a Slugger (before his injury). In other words, he was a Swarmer, who hit like a Slugger. His style vs. Tunney's style also has to be considered in this hypothetical match up.
No doubt Tunney outboxes Sullivan during the fight's duration, though probably not in such one-sided fashion as we think, but I'm not sure he can finish the fight. Even an older Dempsey caught Tunney eventually, and I'm not quite sure Tunney could've taken the ferocity of a prime Dempsey. When it compared to those who came before him, the only heavyweight as ferocious and explosive as Jack was John L.
I do not doubt Sullivan's physical talents ... hand speed, power, chin, stamina, killer instinct, natural strength and pure , true KO power but the man fought in the sport in it's infancy against such a different level talent it is barely the same sport ... under M of Q Tunney , a far more polished and experienced version of Corbett would dice him to pieces.
I saw that interview, I believe McAuliffe was asked during the Carnera-M Baer title bout, if memory serves. He thought Tunney had a better temperament than Corbett, in that Tunney would get less flustered when hit. McAuliffe also felt that Tunney could have succeeded in finishing off a hurt Fitzsimmons. Usually people who , thru the 1910's-1930's, favored Sullivan at #1 were those who saw him in the first half of the 1880's.
I have no idea what to base that on but maybe the trouble Corbett had twice w a prime Sharkey ... it's not a stretch to believe a prime Sullivan at least matched the Sailor in strength, speed , conditioning, chin and power ... Tom had far more M of Q fights against quality opposition .. people also either forget or don't see to know that Sullivan had a decent reach at 74" ... I have always felt Sullivan had skill but never developed it in the manner he would have needed soon after ...
The top punchers who fought Corbett, seem to have caught him eventually. They often looked bad doing it, but his luck ran out eventually. The prime John L Sullivan, who was crushing everybody, would have been no exception.