There is subspace communication that can travel all the way across Federation space much faster than warp travel, and you know that it would have made the front page on the 'Federation news service'.
Curiously, when Jake Sisko becomes a FNS reporter, this seems to represent a major breakthrough for the organization - basically, this kid is their first opening into the secret world of DS9! It wouldn't be difficult to imagine that Starfleet would wish to keep the Dominion thing under wraps for a while, and DS9 would be the ideal location for that.
So a Galaxy
class starship goes missing? Well, it happened in the direction of Bajor, a distant backwater where news really don't travel all that fast... Perhaps she just got delayed, or found a Mysterionite homeworld or something, and will turn up later again?
The interesting thing about the Dominion threat is that it actually got mellowed down soon after the Voyager
left. Contrary to what Sisko was told, there was no major Dominion presence on the Gamma side of the wormhole; there were neither diplomatic nor military overtures to secure passage through the wormhole (apparently because the VR interrogation of Sisko and his crew had revealed the futility of such things); Dominion spies on the Alpha side were either not all that numerous, or weren't engaged in active sabotage after all; and it would not be long before Starfleet vessels gained the ability to defend themselves against phased polaron beams. But right after "Jem'Hadar" and "The Search", the Dominion would be Starfleet's greatest threat...
...And Janeway would just have been told that this threat would be stopped by collapsing the wormhole, by the very man who had already collapsed the wormhole once! So out of all the options available for a journey home, the ones involving the Bajoran wormhole were right out.
Would the EMH be told this much? No particular reason why he should. After all, the ship was not
headed towards a confrontation with the Dominion - this decision had been made early on, long before the EMH became a person.