I think Sullivan would likely perform the best out of those in the early heavyweight era (1860s-1910s), I say likely since we have no film of him. However, stylistically he is a very powerful hitter who is also very durable and insanely tough, he rushes his opponents early on. Compare this to someone like Jeffries who wears his opponent out or Corbett who would try and outbox his opponent. Deontay Wilder is an example of how power is, sometimes, all a fighter needs. While Jeffries and others of the era also had power, Sullivan's bull rushing style seems like it would be much more successful. Not sure if he would be a champion, but I don't think it's improbable for him to achieve success. He's 5'10 and 200 pounds, with ATG power and (apparently) remarkable speed for a man of his size. He has everything a great puncher needs.
John Sullivan In The Modern Era would be a journeyman. But one level below Derek Chisora (gatekeeper). Prime Derek Chisora would beat Prime John Sullivan (by modern rules).
Sullivan was clearly the best talent in the world, and the population of the world was by no means small back then. Training methods might have changed since then, but humans haven't. A standout talent at the point of taking up the sport back then, would be a standout talent taking up the sport today. It is probably fair to assume that if you gave a young John L Sullivan to a modern trainer, they would be able to develop him into some kind of exceptional fighter today.