I was thinking that warnings would be selective, like Sec only warns civs in their territory and Resistance warns civs in their territory - kinda like protecting their own. Also, The Ark isn't some massive country - It's only so big and people are aware of the tension between the 2 factions. It's a good chance that it's common knowledge where some of the "hot spots" of the Ark are, and to stay clear of them.
Considering that any successful combat requires information, if one side made a habit of telling civilians to leave, you can bet that the other side would find a way to intercept the information. Its just a bad idea. Most missions are intended to be carried out with minimal combat anyway, if a firefight breaks out, then something went wrong already.
Look at the container city cutscene. "Let's get in and out before they even know that we are here" and then he freaks out over a single helicopter flying overhead. I should imagine the mass migration of a whole city sector would attract some attention.
Not to mention, where are the civs even gonna go? The Ark is so crowded that the refugees are put 5 families to a crate and then stacked several crates high. As far as the story goes, I can only imagine that the refugees hide in their thick steel crates and hope that they don't get killed the moment they hear gunfire.