Deserted? Deserted what? There is no evidence of desertion. Not every officer or every ship sees combat during war.
But the Enterprise would, obviously, this being Star Trek. And these officers never even mention fighting, yet do express the desire to do something more worthwhile - this being "exploration" rather than defending the Federation against a ruthless juggernaut that was on the verge of killing trillions till the last days of the war.
As for the Dominion, they are mentioned 3 times in the movie.
And every reference is in contradiction with the idea of an ongoing conflict. The Dominion does not negotiate while the war goes on - "Statistical Probabilities" would have put an end to that already. The other two references are to past challenges, associated with lists of other explicitly past challenges. And our heroes are breaking bread with people who during the war were key allies to the Dominion - okay for postwar, out of the question during the war.
There is no evidence the Enterprise and our heroes haven't been involved.
There's direct evidence - Picard laments they have been doing nothing of the sort in living memory, not exploration, not combat, but things directly comparable to "negotiating a territorial dispute at Nevahurd XIIII".
They just happen to be on a diplomatic mission at the moment of the movie.
And are next headed to hover over an archaeological dig on a schedule measured in "seasons". Meanwhile, the rest of Starfleet is supposed to fight the losing battle we see in DS9? While letting Worf have a bit of free time during which he never laments his absence from Sisko's side?
Yeah, there's no indication that Picard didn't report Kirk's (second) death.
No indication that he would have, either. And what's with him hiding the evidence under a pile of rock?
(Not the corpse, obviously, as he could never hope to drag that up the mountain. But he very deliberately abandoned Kirk's badge on that planet rather than taking it with him to be delivered to next of kin. In addition to constructing a private memorial, unnecessary if there was going to be official recognition of the death and subsequently an appropriate non-private memorial.)
Picard would do his duty and give a full report on the Nexus and anything arising from it, Kirk included.
Probably not, considering this would be unusual of him. He's good at keeping secrets, or sharing them with just his bartender rather than with his superiors.
Timo Saloniemi