CV dont really change how games are played, their impact ends up being scouting and since CV vs CV doesnt really work and outside weird misplays (I once saw a Saipan going to the front line, got sunk real fast) each side will typically have the same advantage, if only on opposite flanks.
CVs can deal damage but its not the same as when a Minotaur presents a bit of a side to a USN Cruiser or the wall of skill certain IJN DDs can do, they are on the pathetic side ... heck I faced Enterprise on my Harekaze that sunk me but still took about 4 strikes to do it, CVs outside FDR dont have much alpha short of weird shenanigans like a DD eating a torp thats fairly uncommon, there is some theoretical damage but then again, look at the AP rockets that are so easy to just angle and they end up doing jack shit (maybe outside CLs), in theory they should deal damage but in reality only bots will take damge.
In the end, CVs dont prevent pushes for their damage because about any other class can deal consistently more damage, DDs on the other hand have far bigger alphas and they are as capable as scouts ... sure they have some counters like Radar but then you notice how DD mains when not bitching about CVs are bitching about Radar but DD can also simply DELET BB with a wall of skill as FDR cannot do that, high tier Italian cruisers can do horrible things to cruisers with their high velocity guns with AP (despite the bitching being about SAP, of course they can also do horrible things to DDs with SAP) and then we have the Super Heavy Cruiser known as Thunderer because that is not a BB, not with that LOL PEN guided by His Royal Majesty hand shells, heck Slava despite the Reeeeeeeeeee Russian Bias ended up as a very situational boring ship ... CVs outside low tier simply lack the impact and this is because Tier IV are the training wheels CVs, before you learned that CVs really dont have infinite planes (play Saipan and you learn that very fast, like when sending your 3nd strike out) and you dont get to have the same flight strike 2-3 times and more like 1-2 because AA really ramps up, this is why the seal clubbers are all on the lower tier and why its rare seeing 2 CVs outside Tier IV because Tier VIII USN AA isnt fun for Tier VI planes.
A CV cannot shut a offensive play because its lacks the DAMAGE to do so, it can shut down SOLO attempts because if I see a BB parked on my base, of course I immediate hit f and send whatever I have to stop him ... likely wont sink it as my teammates also have to switch targets and do it for me but anything that gets close to 10km of a CV is very capable of sinking it, once I had a DD closing and I thought it was over but for some reason instead of pushing he decided to a very stupid thing, he turned back, smoked parked in the middle of two small islands when all I had was a Torpedo squadron in the air ... the channel were he stuck himself was only wide enough to allow a torpedo spread, if he pushed I would have to attempt to torpedo him on open waters and I could only do it once before he was within striking range and he knew were I was.
And I am bad at the game and even worst at PvP.