Besides, Leo is a riverboat gambler. He'd rather stay in the running for the big prize, just in case Gussie finds himself looking for a manager again in the near future. Money Leo can always get.
The aftermath of the Series hurt the Yankees even more than it hurt the Cardinals. The firing of Berra left a feeling of distaste among most baseball fans, which the Mets quickly capitalized on by hiring Berra as a coach. (This guy Grant, or whoever is calling the shots for the Mets, is a really good operator.)
With Keane as manager, the Yankees still have problems. As a publicity-getter, Keane isn't going to be the slightest competition for Stengel. Sure, everybody likes him. But once you've disposed of that, you're left an essentially drab, colorless man.
Secondly, if Keane doesn't win the pennant, all the ill feeling about Berra's firing will be reactivated. And even if he does win, he could very well leave the Yankees and go across town to the Mets, where his old friend Bing Devine is now assistant to the president. That would be a perfect ending to the whole crazy episode.