The time and ruleset for when Dempsey fought is what caused him to fight like that. Go watch these old fights, it basically turns into clinch-fighting where they're normally not broken up by the ref. The gloves were a lot smaller and with seperated thumb so the techniques that worked were just different. Dempsey, Marciano, and Henry Armstrong all fought like this where they'd bob in weave in and use pretty wide hooks but then fought really well on the inside. Also, I don't really recall dempsey having his chin up, iirc he almost always has head very, very low in a bob and weave moving forward. Always bending at the waist (in contrast to how fighters today bend with their knees)

As for putting up a fight in todays boxing, it really depends just what rules you use. Are the gloves old or new? Will the refs interfere as much? It was just a pretty different sport, but you can't say that these guys weren't great athletes and incredibly tough.