@ThomasDB is quite right in his comment. It would be difficult to hide, with the modern technology and the press being on top of literally everything.
That doesn't mean the coaches have to be 100% honest or don't have the wiggle-room in press conferences.
For example, Mourinho is quite famous for "overestimating" injuries to his key players, only to figure out later that the player could play just fine (Diego Costa, anyone?). He's far from being the only manager who does tricks like that but perhaps one of the most famous. The main advantage is that you can mess with the opponent managers tactical plans/preparations.
The opposite scenario is also quite common. A key player gets injured in a game, the press asks how serious it is, and the manager might downplay the injury even if he knows that the player might be out for a month or two.
Practically what a manager says about the injuries at the pre-/post-game conferences is not binding in any way, and practically all is fair in this type of mindgame "warfare".
So to answer the OP, why say anything at all? Mind-games! To mess with the morale or tactics of the opponents.
It could also be a matter of man management for his own squad, for instance: i) to cheer up/support the injured player, or ii) to give confidence to a replacement player...