A lot of their missions are probably classified. Waring factions could exploit a new alien weapon (like the Tamarian's weapon that trumped the Enterprise D) and use it against the Federation. I think some missions, like when they save earth, are shared probably to get people to join Starfleet.
Maybe Picard is recognizable, because of the Locutus thing, but do you really think the rest of the crew would be? I mean, I know that the Enterprise has gone on some "notable" missions, but the Federation is pretty big. Does the average Federation citizen pay attention to the "military" a.k.a. Starfleet? I don't think that the average American would have any clue who serves on American aircraft carriers or any other Naval vessel...
Don't forget Admiral Prescott (I think) pointed out Riker's good work against the Borg
. I think that was general knowledge and not something he read before arriving.