There is a bit of a semantic dance that hasn't been addressed. Recently, someone started another thread where they were talking about changing some of the details regarding Greece. One of their points was that there were a couple ships that the Greeks referred to as 'battleships' but were in practice closer to heavy cruiser in size and character with other ships of the period. Why was it called a battleship? Maybe it was because it was the biggest ship in their fleet. I don't actually know, but I think there is an answer.
The word "battleship" is extremely vague. Its only because we have static references, specifically from HoI1 and HoI2, as well as the doctrines of the day from the larger powers that deployed navies, that we all refer to specific ships as "battleships". Tell me; how many countries with coastlines on the Mediterrainian and Atlantic did not have at least one "battleship" by their own definition?
If a country has no battleships, were they at all considered a naval power? If we were to go back to 1938, and ask the experts from Britain, France, Germany, Italy, U.S., Soviet Union and Japan "does a navy need battleships?" what would be their answer?
If the leader of a nation told his people "we're going to take the seas from our enemies" and there were no battleships in their navy, how would the people react?
I suspect battleships are incredibly important. The CV has to earn its role in replacing battleships by having battleships around to be compared to.