There are many options for solving the problem. Military operation is the last one of them. China should be patient and bite its time now. When China's economic/military strength grow to the point that it could more or less match those of the US (note that they don't have to match the US exactly, only get close enough), its influence will automatically cover its own backyard, i.e. the SCS. All the ASEAN nations will be more willing to collaborate with China, like how they are willing to collaborate with the US. With more economic strength, China will have more to offer to the ASEAN nations. And if anything, they will be more willing to give up more since what they have to give will become less and less significant to them when they grow bigger and bigger. Note that China doesn't have to actually own the area. As long as it can get what they need in the area, it should be fine. And they can do that by collaborating with the ASEAN nations in a way that both sides can be happy, or make them an offer they can't refuse
Your assumption that sending CVBG's can solve the problem is solely based on what happened in 1996. In 1996, China did not have the kind of economic influence and military strength that it possesses now, in fact, far from it since they only began their economic development 4 years before that (1992). And now? Almost 20 years have passed and China is no longer the China in 1996. I would go out a limb and assume that China's economic/military strength will grow even bigger in the near future. So in the minds of politicians in Washington, the chance of China backing down again becomes slimmer and slimmer. You have to remember that it is the politicians who make the final decision whether to send in the CVBG, not military commanders. If that's the case, the chance that the US will use the CVBG to intimate China will also become slimmer in the minds of politicians in Washington. In my opinion, in the future, the US will not use its CVBG in the way they did in 1996 unless they have decided that they will go to war with China simply because they know China will not back down again.
Additionally, in 1996, the US sent its CVBG to the Taiwan strait and China stopped firing missiles. that was it. What would happen if China didn't stop firing its missiles? No one knows. Would the US fire missiles at Chinese missile sites, which are located on Chinese mainland and signifies a declaration of an open war with China? Would the US simply continue its normal path through the strait like nothing has happened? Would the US pull out the strait? Would both sides start a war of words, which would last a couple month and end with China stopping military communication again (if there was any back in 1996)? No one knows. So your assumption that sending in the CVBG will end all issues will not always work. It's equally likely that the US might back down and pull out vs. China backing down vs. an open war.
However, what was clear from the incident is that both sides maintained cool during a heated moment and did not let the situation escalate. I would assume both sides will adopt a similar approach in the future in case of another 1996 incident.
You seem to be sure that any conflict in the SCS will escalate. However, history tells us that both the US and China have been normally pretty careful about containing the damage. Korean war and Vietnam war would be two good examples. None of them got out of the initial area where the conflict first started. And whatever interests the SCS has cannot even compare with the strategic importance of Korea and Vietnam to China and the US. Yet, they were still careful not to push the envelop too much.
It's not so much as who started the discussion about the possible war, but your argument is less logical and more on the side of fantasy.