They allow you to project power, as the saying goes, anywhere in the world, for extended durations, without needing the permission or assistance of any friendly nation (i.e. airstrips in a nearby country) [0].

If you're going up against any nation that doesn't have the ability to sink an aircraft carrier (which is almost every nation in the world) they are very far indeed from obsolete.

[0] So long as you have a serious logistics chain and serious support vessels including protective warships; if all you actually have is one aircraft carrier, you're going to find it pretty useless for long-term power projection.