If you think about things for a few moments it kinda becomes obvious, with the costumes especially... Why would a CBS Trek show not be able to use things from another CBS Trek show?
Why would TOS designs be locked up in movie rights?
The obnoxious "updating" of TOS designs has more to do with Kurtzman as an executive producer 1. not respecting the original iconography as previous Star Trek producers had in the past, and 2. wanting to put his mark on the franchise as "the guy who took it from old and boring to fresh and relevant." He probably saw Star Trek Discovery in some ways as "his" Next Generation -- an entry in the franchise that is so fresh and well-produced that it defines the next decade+ of Trek media. And Next Gen had updated designs too!! (Kurtzman neglecting to realize that was because it was set 100 years later than TOS, and TOS designs remained canon...)
Only it didn't work out that way...
Anyways, finally but most importantly: none of the designs *really* matter in making Trek "Real Trek." CBS still could have been making "Real Trek" TV shows, just with different designs. But they made STD instead. So the video is doubly useless IMO...