So in real life, from what I had read the US Navy wasn't interested until very late into the 1980s about going into Black Sea with anything. The transit thru the Dardanelles is very tight and restrictive water way, see this
video of a tanker doing it, sped up footage. Note you can see this video
of a submarine, probably a Type 209 in transit on the surface. Can you really imagine a 688 hull (LA class) or a 637 hull (Sturgeon class) trying to transit thru the area? They would have to surface and with potentially hostile forces along the shoreline with ATGMs or even land based AScMs, if not shore based artillery. Its dangerous and as the engagements off Yemen in the last decade have shown its dangerous to have a ship in range of shore based AScMs. Let alone minefields from either the Turks or the Greeks (or both) in the approaches to the straits?
In addition if you look up how the Turkish Navy is arranged from 1952 thru to even now. They are primarily aimed at their classic enemies of the Greeks and the Arabs on the Eastern Med. So most of their basing is in the Aegean and Eastern Med. There wasn't a feeling that anything in the Black Sea would last longer than an ice cube in the Sahara at high noon in the summer.
That said, the US Navy did do a FON into the Black Sea in 1988. Here is some videos of the Russian/Soviet response to that:
Video 1
Video 2
From most of the Eastern Med is very hard to navigate large naval forces and it isn't very deep water in most locations (averages about 5k ft/1.5km; with the deepest is about 16k ft/4.8km near the waters off Greece) for reference the RMS Titanic is about 12.5k Ft/3.8km down in the Atlantic. From what I have read, the NATO and USN doctrine from the late 70s thru to the 80s; was to allow for the Soviet Black Sea Fleet sortie. Allow for Greece and Turkey fight in and around the Aegean using missile boats and their submarines to whittle down forces. After which what remained would be engaged by the US 6th Fleet, Italian and French Naval Forces arranged in various carrier battle groups (Around the Clemenceau, or CdG in French; and the Garibaldi in Italy). Using Italian and French land based air cover to help protect the fleet and cover the convoys operating in the Med. All while trying to contain the expect Soviet breakout from the Med into either the Indian Ocean or into the Atlantic.
All of which again, means why go into the Black Sea and the tight maneuvering areas of the Aegean, when you have more room even in the Eastern Med, with any sort of fleet?
Finally, remember that the Crimea was where the Soviets had their Black Seas fleet based Naval Aviation units that included long range bombers, such as the Tu-16s, Tu-22s, Tu-22Ms, Tu-95s. In addition to whatever was assigned to the PVO for strike and fighter aircraft. So honestly, why put any surface forces at risk to these missile carrying units that could maximize their load outs and their cycle times (launch, weapons release, recovery) from detection to missile firing?