Coming from a naval point of view, Escorts are like the Fast Attack and Balistic Missle submarines of Star Trek - Small crew, very manueverable, high damage. But despite this smaller crew and much less space for storage and facilities, our deployments are just as long as a carrier with a 20 times larger crew.
And just because a ship is smaller doesn't mean it doesn't have the amenities, they are just wherever they can fit. Life on a submarine is probably very comparable to life on an escort. Crew's Mess doubles as the movie lounge, exercise equipment tucked away in corners of the engine room, things like this. I could see an escort using the holodeck to double as crews mess as well as wardroom.
They mention in several mission dialogues that a specialised ship (mining ship, medical ship etc) will be sent by Starfleet to deal with the situation after you've cleaned up. So I think it's fine; you're there doing casual exploration while dealing with the war. A lot of the missions are system patrols, and you'd want a warship to patrol, wouldn't you?
And the way Starfleet and the galaxy is depicted in the show, there are often only a handful of ships even vaguely near each other at any one time. Often ships are a few days away from each other. So regardless of your role, you need to do what Starfleet does best, stick your nose in everyones business. :p