Snake Island (Zmiyniy Ostrov) was a strategic target because it allows control of portions of continental shelf, entry into Danube and navigation into Odessa. It wasn't useful for supporting of amphibious operations or closing-off of airspace in the area. The island is a rock formation 600m by 400m that had an artificial settlement placed on it for the purpose of acquiring the legal status of an island for the purpose of EEZ claims.
Snake Island can't serve as a staging ground for amphibious operations because it is not large enough and has steep banks. It is also located too far south for any kind of operation other than a limited scenario of creating a separatist region in the Budjak - the part of Ukraine south of Dniester.
Snake Island can't serve as location of medium and long-range air defenses such as S-300 or S-400 because the area allows it to be neutralized by fragmentation from a single Tochka warhead. If the illuminator is neutralized for example by TB2 then unguided FAB-250 bombs can be lobbed by aircraft (Su-24s, Su-27s or even L-39s) from beyond range of Tor with detonation set for airburst at altitude and clear the area.
Russia annexed Crimea in 2014 for a strategic reason. It needed it to properly project power in the Black Sea theater, but even with Crimea it doesn't control the waters and airspace nearly as well as most people imagine.
This is a simple diagram of tactical situation in the air around Crimea:
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Radar horizon for a S-400 radar is denoted by dashed lines. The numbers are altitude and distance of the horizon from the radar. Read it as: radars can't see below altitude of X at distance of Y.
The shaded red circles are maximum ranges of missiles of S-400: 40km for 9M96E, 120km for 9M96E, 250km for 48N6E3 and 400km for 40N6. Only 9M96 and 40N6 have active homing which means that enemy aircraft within ~135km are safe below 1000m (!) in most instances as 40N6 numbers are limited. It essentially leaves the ships to fend for themselves.
This is why Russia tried to use the captured Ukrainian oil rigs as sensor base - with S-400 moved to a further position it improves the situation a little but not enough, as recent attacks demonstrated.
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Such situation requires constant air patrols but Russia doesn't have sufficient number of AWACS aircraft (9 total) and aerial refueling assets (18 Il-78s in Long Range Aviation command) to maintain constant CAP with Flankers which do not carry fuel tanks. This means that Russia has to either wear down its fighters for CAP or scramble them if ground and naval sensors detect the enemy. Radar horizon limits ground-based sensors and for naval radars there's this map of indicative ranges of anti-ship missiles (red) and coastal radars (blue).
Note that these are
not ranges of missiles that Ukraine has but indicative ranges to better understand the problem. Ukraine has/had Neptune which is said to have range of anywhere up to 250km while Harpoon Block I will have up to 140km.
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Black Sea Fleet has few warships with good anti-air capabilities and seaworthiness. It's limited to three Grigorovich frigates (one of them locked out in the Med after straits' closure by Turkey) and Slava. This is why Moskva was such an important asset and why losing it changed everything. Slava was the only mobile and (in theory) survivable S-300 station that with help of other ships could fill the information gap. Once it sunk Russia is back to scrambling jets because while Grigorivich has the same class of air search radar as Moskva it has only Buk which means that it can be effectively shot at by old Ukrainian air-to-surface rockets from beyond the range of missiles. Other Black Sea Fleet ships don't have the seaworthiness or survivability to fill the gap.
This forces Russia to scramble fighters on call but even then the fighters available are not particularly good.
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There are no Su-35S active in Southern Military District. The Su-30SMs have Bars radar with a long maximum theoretical range (indicated on the first map by purple field) that should work well against old Ukrainian jets but the missiles have limited range compared to the range of the radar and in absence of continuous situational awareness that Ukraine has due to NATO information sharing Ukraine can use Su-27s (even with semi active missiles) to effectively engage Su-30SM. Su-27s can be engaged fairly easily. So for Russia it's not a very effective method of maintaining control of airspace at all.
Withdrawing from the island was a practical move because at this point in the war nobody would agree to be sent there.
Ukraine does not need to keep any forces there. Russia does. They need to have boots on the ground to make any claims or present a credible threat. If they don't everyone in the region will treat the island as Ukrainian territory. Ukraine only need someone to plant the flag for propaganda victory.