Interesting how you put that because there were two episodes where we follow Kirk onto the bridge through the turbolift. The camera is moving and could have panned to both sides of the bridge to get the complete view. But in WNMHGB we she the port side and in The Corbimite Maneuver we she the starboard side. That episode (the first series production episode - the one I have listed as episode 3 in production order) shows that each piece of the bridge is distinct and we see each one on screen. In The Doomsday Machine Spock's station has been moved over one, probably to allow camera access. The wedge normally removed to get shots of Spock at his station has been omitted. So the only sections of the bridge not seen in that shot are the omitted one, Communications, and the turbolift. I think it is safe to assume that those two were in their original positions (being connected to the command island) and Spock's station was moved over one and that area left blank for camera access. Just a few minutes later there are a couple of shots that pan across two of the port wedges, the turbolift, communications and to Spock. So through slight of hand, we sort of get a 360 view of the bridge in this episode, we are just one wedge short (the slight of hand).Yes, and here's another thing. I can't remember if it was the book Inside Star Trek (Justman, Solow), or Shatner's book, or possibly somewhere on Ralph Senensky's blog, but I read one of these insiders saying that every director who came on the show wanted to film a scene inside the complete, sealed-up bridge. And they could never make it work.
My thoughts as to the (unstated) reason:
• There would only be enough floor space in there for a hand-held camera, and while they did use one for the barroom brawl in "The Trouble with Tribbles" (in conjunction with the usual big camera), this was years before Steadicam™, and hand-held cameras were too wobbly for anything but brief shots. A full look around would require a long shot with a lot of camera movement— and if you can't do that smoothly enough for the (then) standards of professional television, then the bridge might as well be in pieces, because the audience will never know the difference.
• The heat issue I mentioned earlier might also be a factor. We're told the complete bridge would cook almost like an oven.
But we know the whole bridge was built, and directors found the idea tantalizing.