It occurs to me that if the film had opened with the E in the middle of a (hopefully massive) Dominion battle and the crew and ship obviously exhausted afterward leading into the communication from Dougherty that gets the ball rolling, not only would the film be more exciting but the audience would be given hard evidence of the stresses the Federation was facing. Show the E rescuring people from damaged ships afterward (possibly even a guest star who could oppose Picard's actions later on) and make it clear that this has been going on for weeks/months. The only downsides I can immediately come up with are that it might be a bit reminiscent of the battle at the beginning of FC, and in similar vein might essentially place the most exciting scene of the film at the beginning.
^ Absolutely. While the existing opening scenes of the movie do pay lip service to the Federation being at something of a low ebb due to the ongoing War, they're also a goofy sequence of scenes that are played for comedy value (Picard in a funny hat). IMO it would have been more dramatically satisfying to have seen the effect of the Dominion War on the Federation at some point in the movie. As you say, it might be a little too close to the explosive Borg battle from the start of First Contact, but it'd give the plot some kind of dramatic impetus that it currently lacks, in my opinion.

For me that's what Insurrection is missing. It's maybe 85% of the way there, but the script needed a few more little nips and tucks to just bind things together in a more dramatically pleasing way than what we got.