I've always thought the Captain could use a console or display screen in front of his chair, too, so he can tell ship status with a glance instead of some officer having to bark it out loud to him.
The arms of Picard's chair seemed to display some sort of information, though we never really got a chance to see what. He didn't use them much, but in Encounter at Farpoint, Wesley was able to tell they had a perimeter alert by looking at the readout on the chair arms. Kirk also had controls on the chair, though whether or not they provided any sort of readout I don't know.
The Defiant and Voyager had much larger displays available to the captain. The Defiant had large panels on either side of the chair that seemed to be interfaces just like any other terminal. And Voyager had a monitor of some sort that folded down when not in use, but were available when the captain or first officer needed it.
So I guess there is some information available directly to the captain.