Cheap drones are pretty useless against large naval vessels. Making a dent in those ships requires a heavy, specialized penetrating warheads. And even then you'll need to score several hits.
Just the warhead alone on a standard anti-ship weapon weighs more than an entire Shahed-136 drone.
https://en.wikipedia.org/wiki/1967_USS_Forrestal_fire
(~20kg warhead)
Imagine trying to launch fighters when there are explosions on the deck from swarms of drones. And of course the fighters themselves could be hit and destroyed. An aircraft carrier that can't launch fighters is pretty much worthless.
I disagree: lots of cheap drones would be extremely effective against an aircraft carrier. They don't need to sink the ship; they just need to damage the jets or disrupt operations on the flight deck. Even a small drone is a serious threat to a jet. How can a carrier defend against a drone swarm? They only have so much ammunition for those CWIS guns, and defending against the swarm will probably cost a lot more than the swarm itself does.
Of course, this assumes the carrier is within range of the drone swarm, but that seems to be the assumption in this line of argument.
Eventually, I think they'll have more cost-effective defenses against small, cheap drones, but they don't have them yet.
You could fly an FPV drone into the hangar and smash a plane full of ordinance if you get lucky.
Unfortunately warships have a lot of flammables and explosives aboard.
To sink it, yes.
To render it useless for a while is easier.
https://en.wikipedia.org/wiki/USS_Enterprise_fire
All from a little drone-sized warhead.