In air refueling lets counties drop bombs on the other side of the planet. Aircraft carriers are cheaper but that assumes they are not sitting ducks and we can't find a friendly airfield somewhere.
An carrier on the other hand is a thousand foot runway that can move at 56 km/hr.
however, airfields are also vastly cheaper to build and maintain, and if they do get "destroyed" they can be rebuilt or replaced (possibly elsewhere) pretty quickly. in comparison to hitting a multi-billion dollar carrier so hard it's destroyed or taken out of service. In a hot war between China and the US, I'd bet China could rebuild/replace airfields faster than the US can rebuild/replace their carriers.
On a related note, I submit that this very issue is why the US and its allies are so against the Chinese trying to build up those little artificial islands, harbors and airfields, in those disputed waters. Not only does it allow China to project force out to greater ranges, with more overlap, and gives them more options in a shooting war, it also gives them more experience in building them up, and learning how to do it better and faster.
And it sends a signal to all the nations in the region, at least as an implied threat. It tells all the nearby nations, and the US, to think very very carefully about whether they want to get in a shooting war with China, whether over Taiwan or Korea, etc. Because while the US might have some very powerful toys in the theater, very modern and sophisticated, they are also very very expensive, and far from home, unlike the Chinese forces, and can easily be outnumbered. They can be attacked from increasingly shorter ranges, and whatever losses the Chinese do take, their losses will be cheaper and faster to replace than the American losses in theater.
So sure, if the enemy can't easily sink all your aircraft carriers then they are a great option. Sadly, I don't think that's anywhere close to proven.
Based on practitioners and experts I've read, finding ships is very difficult. The ocean is a huge place; remember there's far more ocean than land area. Imagine if you were told that somewhere in North America there's an object the size of an aircraft carrier, and it's moving around. How hard would it be to find it? The Pacific is much larger (though if there's a war over a contested location, the search area becomes much smaller).
Also, there are not nearly enough resources to watch the whole ocean at once. The U.S. military can't even monitor all of Afghanistan, or even all of contested regions in Afghanistan.
There is also having a sore butt, sore back, and no place to poo.
Other than that, a "drone mode" would mostly work. Sleep would be pretty uncomfortable, but might work OK enough.
That said, the capability ought to exist. Even if never used in war, the mere threat of the capability is beneficial. It's also usable to move aircraft around, for aircraft sales and to move aircraft closer to an expected future battle.
This is probably incredibly naïve but wouldn't it help us get a "friendly" airfield if we could say something like "you can allow us to use your airfield or we will move in our ships in N hours anyway". Am I missing something here?
When things get serious, thoughts like "I would never do such a thing!" are quickly cast aside.
I definitely feel that played a part in allowing Germany in WW2
One announced U.S. strategy in the Pacific is to spread forces widely rather than concentrate them, and to utilize bases of friendly nations (who don't need U.S. coercion these days with the Chinese government coercing them right into America's arms). The U.S. recently made such agreements with Australia, The Philippines, Vietnam, and Singapore, and of course has long-standing bases in Japan and South Korea. Indonesia is almost the only major nation left in the region who isn't on board.
Yes, that friendly airfield assumption is one main point of the aircraft carrier.