Both men’s careers skyrocketed after beating the superstar of their eras, who was more known among the general public in their respective eras?
Well he did have the advantage of having the technology to be able to broadcast his fights at all to be fair, but was he more culturally relevant in the 20s after beating Dempsey than Corbett was in the 1890s after beating Sullivan?
Corbett's fame lasted longer IMO, he milked it until the loss to Fitz for 5 years and drew it out until 1903, losing to Jeff and even afterwards with theatrical performances, his fame was limited to the English speaking world but so to, I think, was Tunney's. Both were Irish-American and as an Irishman I can attest Corbett is the most famous in Ireland right now even though he preceded Gene by 35 odd years.
It kind of depends what you mean. Tunney got the biggest purse in the history of the sport, up to that point and for some time after, for his rematch with Dempsey. The uncharitable might say that this was because he was fighting Dempsey. Corbett fell into the usual trap of champions of that era, of making money without putting their title on the line, and he was coining it. Both were masters of rewriting history in their favor. Both were overrepresented on top ten lists, or top twenty lists, long after they retired,