I did already educate myself some months ago of carrier air wing's supposed capacity at surrounding the carrier strike group with greater strategic depth against older destroyers to maintain dominance in modern high seas (hi-hi-hi mission by a Bug with no attempts for stealth within radar horizon and clutter yielded 1000-km in militarized calculations).
Some illu.:


There were many historical cat-and-mouse games played between a US aircraft carrier accompanied by only a single Aegis destroyer (the same game with the same risky group arrangement was played on Iran during no-fly zone patrols in OEF and OIF) against packs of unaided Chinese and Russian Sovremmeny-class destroyers roaming the Pacific between 1992~2000 to test aircraft carrier's independent readiness for unscheduled meeting with hostile ships in open and closed ocean, eventually resulting in massive humiliation on those technologically inferior destroyers' part in their constant failure to gain a safe standoff strike opportunity against the aircraft carrier-Aegis ship duo. The obvious outcome of those incidents was the losers of those games spending the next decade trying to bridge the very visible 'range' gap between a carrier's aircraft-endowed strike and a destroyer's entirely cruise missile-endowed strike, which was their navies' only call on first-striking the greatest stakeholder of American naval might of the day because they couldn't catch up to the US in terms of purely naval aviation strength (couldn't maintain carrier forces that could match America's), using ten to twenty years of new advancement in their technology since their last attempts.
They clearly saw that there
were ways for missiles - ramjet design and solid-booster missiles, or a heavier missile with heavier engine - to outfly an aircraft and strike the carrier first before the aircraft hit them, but there was a problem that they couldn't fix at that time: inferior precision. Missiles launched 1000km away from the ship itself couldn't match the precision of a missile launched by an aircraft 200km away, almost to a degree that they could never hit the carrier, that they couldn't even attempt. The TASM of the US Navy that still is the longest-ranging dedicated anti-ship cruise missile ever put into service was decommissioned for the same fathomed reason: inadequate precision and inflexible mid-course mobility at far enough distance. Now there were very obvious attempts by other nations who face the same danger from US aircraft carriers to find means to fight against them from where the US left off; maintain the range, and increase precision, speed, and mobility through better engines, inertial guidance, GPS, active-radar seekers, and symmetric improvement of the present guidance systems of AShM so that they can attack an aircraft carrier and run away before its air wing's full power can be unleashed on them; it consolidates more than a decade of research and developmental efforts.
The illustration some posts up (the one with two supersonic missiles dropping down on a carrier) drawn by an American as far as I know one or two years ago cited only by American papers procured by DoD and Congress seek to visually express the discouraging scenario of American aircraft carriers dangerously exposed to modern-day standoff DFA threat from the right Chinese and Russian missiles equipped in the right destroyers that with no doubt will be put into service some day. The new defense against the new threats that were intended to nullify the aircraft carriers' old defense - the 'area denial' defense through greater strategic depth and greater surveillance opportunity in both time and distance - steadily shifted its facilitator from the aircraft carriers themselves to Aegis ships because of the latter's less obvious weakness against standoff missiles. Aircraft carriers will continue to keep their undisputed title as the foremost instrument of naval firepower superiority, fear and force projection, but will continue to develop a critical vulnerability - one that Aegis ships won't develop any further than they did until now - against new asymmetrically advanced adversaries that needs to be rectified by continuing to expand its level of companionship and integration with the strongest Aegis missile-defense warships to sail alongside them. The great vulnerability of aircraft carriers against longer-range standoff missiles is not the same one that Aegis ships share, as the nature and level of threat still remain the same to them, and its expansion in complexity serves to close the gap between a lone aircraft carrier and a lone Aegis ship as to which one will be the safer one to sail on in seas occupied by hostile forces.
Within the NATO, there's only one anti-ship cruise missile in existence that's designed to challenge the power of a 60,000+ tonnage non-American aircraft carrier in the high seas in sole combat from a standoff distance - anti-ship variant Hyunmoo-IIIC with 750-km range, a Korean reverse-engineering manifestation of the preserved American TASM project. Hyunmoo-IIIC was a crucially necessary angel to allow a destroyer-and-submarine-only strike group of the ROKN to deter and defend against a carrier strike group of its three great power neighbors in NEA in the probable events that ROKN due to some circumstances were not able to avail the US 7th Fleet's help. Thirty-six of those TASM-derived missiles will be installed aboard each KDX-III to facilitate the first modern toe-to-toe battle between destroyers and carriers that will not see the destroyer squad retreat in fear of the carrier's indefensible first-strike, under conditions that the KDX-III's long-range attack capability was maintained at a similar or greater level to aircraft carriers armed with Mig-29 and Su-33.
Admittedly Russian, Chinese and Japanese carriers will be far weaker than the Nimitz-class and the vaunted Gerald Ford-class supercarriers, but despite their comparative weakness there's no other equipment except the strongest-built Aegis ship or the quietest AIP submarine armed with the strongest possible anti-ship armaments that we can bet on to develop our own indigenous defense and deterrence capability against an naval force that possess an aircraft carrier. KDX-III's growth as the largest and strongest surface warship to contain an Aegis combat system aboard its superstructure is a slow but inexorable progress that we fully plan to sustain for many years to come in preparation for the day that ROKN's destroyers as lonewolves will have to fight or at least try to fight a carrier strike group of any belligerents, because the alternative will not be acceptable.
Tbh, aircraft carriers and Aegis ships do not always perform common duties, so we'll just say Nimitz and KDX-III perform the best among ships of the same class in their respective duties.
P.S.:The 11,000-ton 'cruiser-christened-destroyer' KDX-III does have more advanced C4I and EW suites than Ticonderoga-class cruisers and
obliged to pack greater firepower because of strategic necessity that was fulfilled. Stealth and IR reduction applied on KDX-III that under no circumstances can be retro-applied on Ticonderoga gives it a raw radar profile equivalent to a corvette in defiance of the KDX-III's 20% greater physical size in comparison to the 9,600-ton American ship; it was already confirmed from where I was two years ago by everyone who had access to and examined the ROC of the KDX-III and the underlying reasons for the ROC.