Then again, none of the Enterprises, least of all the E-E, was considered the Starfleet Flagship.
The E-D used to be the Federation Flagship, whatever that means; doesn't sound particularly military. The E-E a year after her launch was considered an "advanced" vessel, at the end of her one-year shakedown. So possibly a testbed for buggy systems, always undergoing maintenance?
How the Breen got to Earth is an interesting question. Starships probably played little role in the actual fight over Earth, though, as generally very few are anywhere near Earth even during crises, and planetary defenses tend to make short work of thousands of ships (unless shut down by superbeings or secret codes or whatnot). But if the Breen simply fought their way to Earth through dozens of defended lightyears by firing their energy damper gun, one would expect starships to rush in to defend Earth, too.
Timo Saloniemi