Now that IS fascinating! All those years staring at these episodes -- and I never noticed that the triangles were never applied to the "witness stand" chair.
At STAR TREK CONTINUES, we used our Burke 116 "Captain's Chair" that usually has a home (and black pads) in the Captain's Quarters or Briefing Room. Ralph Miller made a set of the checkerboard pads from the fabric we had woven.
Looking back at TREK, the 116 chairs usually DID have triangles -- such as in "The Menagerie"
Interesting to note that they just grabbed a couple of 115 "no arms" chairs for the ending shot in Part One of "Menagerie." If you look closely, you'll see that Kirk and Mendez chairs are missing their arms!
The Checkerboard Captain's Chair generally shows up in a place other than Kirk's quarters -- such as in "Amok TIme" in Spock's quarters...
Note both the 115 and 116 Burke covered in Checkerboard fabric in Yeoman's Rands quarters -- early in the first season episode "The Enemy Within"...
And the Checkerboard fabric shows up in guest quarters several times -- including here in Sarek and Amanda's shipboard accommodations on "Journey to Babel"