"Super destroyers" were they worth it?

Then more like an enlarged Gearing or Sumner? Probably could do it on around 3,000 tons. For reference see:
Allen M. Sumner class destroyer - Wikipedia, the free encyclopedia
Gearing class destroyer - Wikipedia, the free encyclopedia

 

What you get looks very much like an Akitsuki, the Japanese twin 100mm mount was not much smaller than the US 5" twin, or a Porter or Mogador with actually functioning DP guns. WW2 destroyers cannot generally survive a a single torpedo hit, and even at 3000t that's still going to be true, but a WW2 fleet requires a "screen" to protect the big boys and that screen is going to take losses, so making then too big and expensive is not a great idea.

 

Still, I think such a ship would need light armor at least to protect AA directors and fire control positions from strafing by lights cannons and MGs of enemy planes, and probably the same armor to protect boilers, turbines and shell rooms.

Regards,