I'm obviously missing some facts about the region, as per the OP's hypothesis, because a naive look at the map suggests that even without Crimea, Russia already has coastline onto the Black Sea between its borders with Ukraine and Georgia... there must be a strong reason in favour of invading another country rather than developing ports on the land they already have, if the goal is simply "control over the Black Sea and trade access to the Mediterranean Sea".
As per the video https://youtu.be/If61baWF4GE), above and beyond the geopolitical reasons for Russia's invasion of Ukraine, the main motive is economical:
1. Russia is in effect a petrostate; it lives and dies by its exports of natural gas and oil.
2. Immense fields of underwater natural gas and oil were discovered in 2012 in the Crimea maritime exclusive economic zone.
3. If Crimea stays with Ukraine, it lets Ukraine become a new petrostate and challenge Russia (take over its European customers), which to Russia is an existential threat. Hence the annexation of Crimea in 2014: not only for Sevastopol and access to the Mediterranean, but to exploit the oil fields (and/or prevent Ukraine from exploiting them).
4. But the supply of 80% of the fresh water to Crimea comes from a canal [0] that connects it to the Dnieper river in Ukraine; after the annexation, Ukraine shut the canal down, effectively thirsting Crimea to death.
5. If Russia wants to stay in Crimea (which it believes is necessary for its own survival), it needs to reopen that canal, and therefore control at least the southern part of Ukraine and possibly its whole Mediterranean coastline.
(It's not just about the canal, but the canal is a very big part of it.)
Russia's Black Sea coast is shallow and lacks large, sheltered harbours. It would cost billions and takes decades to build a proper port there.
Russia had actually be leasing the port at Sevastopol from the Ukraine since the fall of the USSR and had their Mediterranean fleet based there even before the invasion in 2014. They had a lot to lose in an anti-Russian Ukrainian government.
