Well yes you are basically right.

For sinking other ships a carrier generally is your best bet, provided its large and fast enough to actually be useful. A carrier's airwing can deliver a higher weight in ordnance over a much further range more quickly than a battleship could ever hope to.

But carriers also have their limits. For one is that until late war air operations stopped when the sun went down. Meaning if you wanted to hit stuff under cover of darkness it had to be from a ship, and the battleship thus was the biggest baddest thing around to do that.

The efforts of the USN and IJN around Guadalcanal in the Fall of 1942 are textbook for both fleets trying to fit what assets they had on hand to complete the mission. And that both were willing to commit battleships to surface action in a confined area accepting the risks of superior training and torpedo usage by the IJN, and the threat of a dawn air strike in retaliation by the USN/USMC.

We can also look at their employment as AA platforms. Sure a destroyer might be faster(sort of neglible when it comes to the USN fast battleships) but that small size and speed also limits armaments. A Fletcher class DD would have 5 single mounts for her 5in dual purpose guns which were some of the most effective heavy AA ever used. An Iowa class BB would have 4 times that many! With similar relationships in numbers of lighter 40mm and 20mm guns.

As a practical example the USS South Dakota was part of the escort of the Enterprise at the battle of the Santa Cruz Islands, trailing about 1k yards behind the carrier. Because she was such a large ship she attracted some attackers of her own, and many Japanese aircraft sought to escape past her through the gaps in the screen to either side. In turn she put out so much flak it was described variously as the 4th of July fireworks, or a Christmas tree all lit up, and Enterprise even sent a query on if she was damaged and on fire. She is credited with about 2 dozen enemy aircraft from the battle, quite the tally even if its probably a bit over stated, consider the IJN only lost about 100 aircraft the whole day. For all that she suffered just a single bomb hit which was shrugged off and was far more damaged in a collision a few days later than the battle.

So we can see that the battleship still found its niche as a part of a task force, either asserting tis weight where a carrier couldn't, or acting as the bulwark of its defense. And each navy answered the changing times differently, the IJN being notable for how hard they tried to set up a surface engagement for their battle line essentially all the way through the war.