Despite the Enterprise-D (then the E) was the prestigious flagship, her crew members didn't ever seem to be having a reputation of exceptional heroes. Picard was a well-respected Captain as Kirk during TOS era.
Look at Captain Jellico, he treated the Enterprise officers like typical Starfleet officers.
In Unfication, the bar singer didn't immediately recognize Riker as the Enterprise XO. She saw the uniform, then realized by talking with him he was from the ship that killed her former husband. When they wear civilian clothes on a bar or Risa, they are incognito.