IMHO, Pete Rose was allowed to keep playing (and managing) by the Reds, for the publicity. He was a gate attraction, and they loved having their hometown hero come back and chase Ty Cobb.
If the Reds didn't want Rose in the mid 80s, do you think he would have been kept around long enough to eclipse Cobb? Did the fact that he was the manager, and kept putting his name in the lineup, a factor? If the Reds had another manager, would Pete have gotten less playing time?
IMHO, I think the Yankees value winning too much, and won't keep Jeter around just to chase the record. Also, I don't see Jeter playing for another team, especially a small-market team that would want to bring him on, to get an attendance boost.