What is important about the battle of Kharkiv is what it shows about Ukrainian military behaviour compared to what the battle of Donbass shows about the Russian army.
What has Ukraine been doing around Kharkiv in recent days? It seems to have set a clear objective (to keep the Russians away from the city to protect it and give options for future operations). They have identified a weak part of the Russian lines (which were focused on the Donbass). The forces gathered were appropriate for the operation. They waited until they were ready and methodically took all the right towns to reach their goal. In 4 days they achieved their objective, the Russians seemed surprised and unable to react at all.
Compare this to the battle of the Donbass. Stinking with unprofessionalism and failure. The Russians essentially telegraph the operation to the Ukrainians (you know we are coming). They then rush seemingly haphazard forces into the theatre. Many were exhausted from Kiev, others had no idea what they were doing. Their supply lines were well exposed to attack. When they attacked, it was before they were ready, they didn't have enough strength. Instead, they seemed to engage their forces in small groups. They have taken almost no territory; suffered huge losses and have not moved for almost a week. Their logistics are still uneven, their soldiers show great hesitation, and their attacks seem uncoordinated. Now they are preparing to launch exhausted troops from Mariupol to attack from the south.
Let's summarise. One side is professional, setting important and achievable goals and concentrating the force necessary to achieve them. The other is pompous, unprofessional, attacks when it is not ready and wastes its own strength. When you look at the facts from this perspective, it is not difficult to understand how the war has developed so far.
The strike on the island, that explains the strikes on the air defence systems. Suppression followed by a real air strike, unstoppable as the SAM picket is at the bottom of the water (Moskva) and the land-based systems are destroyed. As it is far to the West, there is little or no air cover.
As for the reason for this relentlessness, the island has a view of the entire western basin of the Black Sea
Its neutralisation would allow the passage of ships from Romania to the south-west Ukraine
And it is a symbol.