Short lecture time ho!
A Flight IIA Burke is basically an amalgamation of the WW2 DE, DD, CL, and CA. It does the roles of all those ships, and calling it a destroyer is still accurate today because it is the workhorse of the fleet, and destroyers were the workhorse of the fleet, and I love destroyers and Arleigh Burke loved destroyers so there. (Also the US has more Burkes than many nations have
ships.) Now, in terms of ASuW and ASW, the Flight IIA Burkes are less capable than the original Flight I Burkes, because they lost the two quadpack Harpoon launchers and the towed array sonar to fit in the helo hangar (otoh Romeo Seahawks are ASW hotness so that's been mitigated somewhat). As a result, a Burke's only missile options for ASuW are to use the SM-2 and SM-6 Standard Missiles as short range ad hoc antiship missiles. This is not
as bad as it sounds given that 1) modern ships have jack and shit for armor and 2) a one ton missile hitting you at mach 2 has a shitload of kinetic energy, and 3) unlike every other Navy in the world, the USN's primary antiship platform is a carrier-based fighter. Carriers launch Hornets to kill ships, Burkes put up a wall of SAMs to guard the carrier.
As Break said, this is changing, given the USN's pivot to asia, and with a number of AShMs in competition to be selected to replace Harpoon (Block IV Tomahawk, LRASM, NSM, modified Harpoon).
(The Japanese and Korean Aegis DDGs retain their helo hangars and AShMs by going "fuckit" and lengthening the hull. The Korean Sejong the Great-class has 128 VLS cells and 16 Korean Not!Harpoons.)
Everything I said wrt Burkes applies to the Ticos, which were literally Spruance-class DDs with Aegis, but I'd add that the CL or CA equivalency is stronger with the Ticos because 1) larger ship and more weapons, and 2) flagship fittings for a Commodore. Ticos serve as flotilla leaders, shepherding Burkes, so that should put them into CL or CA equivalence. The same argument can be made for the Japanese Atago-class DDGs, as they're 2 decks higher than Burkes because of the flagship fittings to be flotilla leaders.
Those would likely nuke the island bases humanity will have to rebuild post-war. That said, Abyssals mess with modern tracking systems so much you practically have to take manual control of each missile, while dodging incoming shellfire and torpedoes.
Can we not be talking about casually throwing nukes around, ayy.
As a general rule, nuking a place that you intend to
capture is not a good idea.