I would have loved to have seen a more fitting introduction to the new Enterprise, she is one of the characters and deserved better.
A great storming entry into battle would have been excellent; start the film with the Borg attack or approach on Earth and let the first big money shot of the film involve the Enterprise cleaning house.
Interestingly enough an earlier draft of First Contact did start that way.
Also it had the Federation fighting a Borg fleet
and holding their own due to new technology and weapons (quantum torpedoes were apparently way more impressive anti-Borg weapons in this version instead of just being fancier photon torpedoes and the Enterprise-E was using some kind of Borg resistant armor instead of regular shields).