Curiously, I don't recall Picard ever acting as you describe, but I can recall Sisko doing that on several occasions. Maybe not the firing phasers part, but I think we saw Sisko working in Main Engneering on the Defiant more times than we saw Picard in the room on the Enterprise.
Which you'd expect as the Enterprise had a huge crew, and the Defiant would often be worked by a crew of 20 to 30.