Actually they don't for the first two anymore.
The Stories of CVNs outpacing their escorts came from the time when Steam power ships was still common in the fleet. Those had terrible acceleration due to the need to baby the boilers lest you risk an explosion. A nuke ship is far more able to raise pressure then oil burners were. But there is still a decent lag team between the throttle man opening up the throttle and the ship picking up speed.
But when the Spruances hit the fleet, they routinely out accelerated the Carriers cause you can just slammed the throttle forward and have the FULL POWER of the turbines immediately to the props. With those often need to feather their propellers, both them the ticos and the Burkes have adjustable props, to allow the carrier to catch up.
This is still true for this day for the Burkes. The only time they get left behind in acceration is when the sea state is bad enough to force them to limit themselves. Which the CVN being F U size don't care bout, and those Sea states are on the side of being uncommon in good weather.
And a few years back the Nimitz's started to outright fail their speed trials due to all the weight they put on from the gear. The Ford manage to bring it back to snuff but even so she still gets out done by the Burkes in acceration.
Like no really the Burkes have HALF THE HORSEPOWER OF THE CVNS on a hull a not even a tenth of the size. They are extremely speedy little things.
Endurance is the only one the CVN can claim these days. And only a soon to be court martial fool outruns his screen.