In my view, the biggest problem with CV really stems from its real life source. Quick history recap: arrival of CVs turned naval fleet balance on its head. If you look at the naval tactics and squadron composition, it really splits up into "before CV" and "after CV". No other ship class has had such a severe impact on how naval battles are fought. Why? Damage projection and battlefield control. While many ship classes have toyed with ability to launch planes, to varying degrees of success, CVs took the idea and cranked it up to 11. Having planes allows the CV unprecedented presence on the battlefield and reach distance. Up till then battleships were the kings of the high seas, having heavy firepower and extensive range of 15-25 nmi. Enter CVs, with plane range of over a hundred nmi, easily. That's an order of magnitude more! CV is the main reason to why BBs are extinct or limited to being (sometimes) floating museums.
This very same reason makes CVs so potent in this game as well. CV has the absolutely unbeatable ability to see anywhere and strike anywhere, and do so without any significant threat to itself. If focused a CV can easily shut down and destroy any ship anywhere on the map. The only ship class capable of countering the CV without relying on team support is another CV. All other ships must rely on teamwork to fend the CV off. And how much teamwork do you actually see in your typical random match?
This disparity and superiority of CVs over any other ship class is why we can only have one or two of them per team, and from there stems the other issue. It's not about the skill ceiling or floor, its about the case when a below-average CV player meets a good CV player and gets absolutely dominated. There are no other CVs on their team to pick up the slack, which results in the better player easily achieving air superiority and ability to project their firepower onto any enemy of their choosing nearly unopposed.
So, basically, the problem with CVs is that you need another CV to keep the enemy in check. And if the skill gap between players is too great (or even just RNG gods prefering one over the other), you end up with a greatly superior ship running free doing whatever they want to.
There really is no easy solution to this problem. In a more or less rock-paper-scissors balance of other classes CV is the shotgun. The most reasonable solution would be scrapping the class as a whole and returning it to the drawing table. Or not returning it at all. Making the CV fun to play as and against will require a lot of effort from the developers. And I'm pretty sure they are aware of this.