This sounds almost exactly like what I remember in Decatur, Alabama, on the Telecable system there. My father was good friends with the manager there, and I got the chance to visit the studios there when I was a kid back in the late 1970s. From what I recall, the device was about a waist-high console (with the motors and equipment inside) with the top containing a semi-circle of weather instruments along a curved back (brown, which showed up as gray on the screen). A rotating mirror-like (about the size of a large ladies' compact) camera situated toward the front of the machine went back and forth about 120 degrees or so to shoot the continuous panorama of dials, about six or seven in total, IIRC; I don't remember for certain whether there was a clock among them. To prove to my skeptical mind that it was in fact a small TV camera, Dad placed his thumb in front and, as I was watching the monitor nearby, I saw that he covered up one of the dials! I wonder to this day what some casual viewer flipping over to channel 11 (I think) to check out the temp outside at that moment must have thought ...
The background music was, like what you remember in Savannah, an easy listening station, WRSA-FM in Huntsville, which is now adult contemporary in format, with the disappearance of Muzak from American life.
I wish I could help you with a video clip of that, but we didn't have a VCR back then, and probably would have considered it silly to waste tape on something like that if we did. I suspect you're going to come up zero on YouTube also, for much the same reasons elsewhere.
My questions are twofold: 1) how widespread was the B&W dial-display weather elsewhere in the county, and 2) does anybody know what the device was called? Probably something like "Temp-Matic" or "Weather View," perhaps.