The thought does go through my mind that perhaps there is a danger of expecting the opposition to do as we would do?
Are the Chinese going to be terribly bothered if there is a degree of friendly collateral damage when they take out a US carrier?
That depends on a lot of factors, including what is meant by "friendly." For example, what if the ship sunk instead of the American carrier is Russian? Or British? Or Japanese? or Australian? or S. Korean? Are they considered friendly and will they be "terribly bothered"? It seems to me that this is a complicated problem involving more than just the USA and China. But I could be wrong and welcome other viewpoints.
Another consideration is the reaction of American forces in the region if the carrier dodges the missile, but the missile sinks a ship friendly to America, like Britain, Japan, Taiwan, Australia, S Korea, etc etc. Will the Americans be "terribly bothered" and its forces react in some hostile manner? Will America's allies in the region, like Japan, S. Korea, Taiwan, etc join America in a forceful military response to China's provocative action? I believe these considerations should be included in the equation. But again I could be wrong and welcome other viewpoints.
To my mind, this is a very high risk operation with very VERY high stakes. Even assuming China used ASBMs to successfully sink all of the American carriers, America retains the ability to visit a lot of hurt on China using several other non-nuclear weapon systems not dependent on carriers. And that does not include America's allies in the region. So to my mind this is about a lot more than the ability to sink CVNs using ASBMs. But I could be wrong and I'm open to have my mind changed.