I confess that I wasn't able to watch the video ... I live on a farm and our DSL internet running on old rural phone lines won't support video. With that said, any serious decimation of factual information is nearly always by written text with supporting references.
The crew size for a Nimitz class carrier is overstated by about 1,000 in your post.
There are ten Nimitz class carriers in the U.S. fleet. Five are stationed on the east coast (Norfolk, VA and Newport News, VA), four on the west coast (Bremerton, WA and San Diego, CA, the Carl Vinson is based in San Diego), and one in Yokosuka, Japan.
https://www.navy.mil/Resources/Fact-Files/Display-FactFiles/Article/2169795/aircraft-carriers-cvnGiven the location of the Black Sea, and the only entrance into it being from the Mediterranean, it would seem to make way more sense to send a ship from a port on the U.S. east coast rather than one from a west coast port. Even by sailing through the Panama Canal (and I wonder if a Nimitz class carrier would fit), sailing from Virginia is a shorter route to the Bosporus Strait than sailing from San Diego.