A better question would be, why would they? Stafleet has hundreds, if not thousands, of starships. The Enterprise may be special, but she's not the center of the universe -- quite frankly, for the course of the war, DS9 itself was far more important than the Enterprise. Starfleet could afford to lose one starship; it couldn't afford to lose the space station that controlled access to the Wormhole.
You like answering questions with questions.
Because I'm trying to suggest things that the alternate questions imply.
The Enterprise is not the center of the universe, that too is obvious, however she was important. She and her sister ship were the two most advanced and powerful ships in the fleet.
Or, rather, they were the two most advanced and powerful ships in the fleet when they went on active duty in 2372. The Dominion War didn't break out until the very end of 2373, and lasted until almost the beginning of 2376. And VOY established pretty firmly that the
Prometheus class, introduced in 2374, was the most tactically advanced ship in the fleet. So, the
Enterprise clearly wasn't the biggest fish in the pond anymore.
Secondly, so what? That the
Sovereign class ships are very powerful does not mean that they would necessarily be around the area of space that
Deep Space Nine was set in. Remember, the two areas we saw most frequently were Bajor, Cardassia, and the space between those two worlds. But the Dominion War was a
massive war that was being fought
all over the place. Just like there's no particular reason for soldiers fighting on the beaches of Normandy to be talking about what the aircraft carrier
Enterprise was doing if she was in the Pacific Ocean, there's no particular reason for soldiers on Deep Space 9 to be prattling on about what the starship
Enterprise was doing if she was, say, fighting in along the Benzite front, or near the Romulan border to defend against Dominion incursions over the Neutral Zone (mentioned in "In the Pale Moonlight"), or if she was attached to the Sol Sector. The war was
big, and there's no particular reason for the
Enterprise to be a topic of conversation at DS9 other than that Worf used to serve there.
Their power and rarity would also mean that their whereabouts and readiness would be of strategic importance to Starfleet and its overall planning for the War.
Yeah, but we only ever saw a small part of the planning on Starfleet's part: We saw the planning that went on in the Bajor/Cardassian sector hotspots. We never saw what kind of planning went into defending the core Federation worlds like Earth or Vulcan or Andor. We never saw who was fighting -- and lost -- for Betazed. We never saw who fought off the Cardassian raiders who were crossing the Romulan Neutral Zone. We never saw who chased off the Breen from Earth. Etc. So the fact that we didn't see them mention the
Enterprise on DS9 or at Starbase 375 means very little.
And it did seem like Starfleet was willing to give up control of DS9, but not the minefield that prevented its use. They were going to leave it as long as they could, and it took a bit of convincing to get the Admirals to go along with Sisko's plan.
Yeah, because they were being bloody stupid. DS9 was, in Sisko's words, the most important piece of real estate in the Alpha Quadrant -- and he obviously managed to convince them of that.
Now to go back and answer your other questions, Saving Private Ryan did not take place in one of the planning areas of the War, DS9 did.
Actually, it did. We saw
plenty of planning sequences after the Allies had taken control of the beaches in that film. But, as I noted above, World War II was a pretty big war; even given the legendary status of the USS
Enterprise, there's no particular reason for soldiers in Occupied France to be talking about it. They had their own part of the war to fight.
Casablanca is, at its heart, a love story that also includes an espionage and escape plot, so there too there is no need to mention Churchill.
Exactly.