Alternately, we send in the aircraft carrier, it is destroyed, what then? We nuke a population center? Unlikely. Without missile defense, we are powerless. Kind of like not having air support during WWII.
Check Mate.
That wouldn't need to be nuclear. We still have rather a few submarines for use against capital ships, invasion ships, etc. If they can do it, we can:
http://www.dailymail.co.uk/news/article-492804/The-uninvited...
Sadly, the very point of a stealthy submarine makes it useless for what a carrier group does very well - signify military presence and imply the willingness to use it.
Of course, carrier groups often include submarines. Perhaps in the future, carrier groups will have more submarines to get around the ballistic missile threat. Take out the carrier, and you've just blasted off the visible part of the iceberg, no more.
Also, spreading out thousands of crewmembers on a swarm of small surface craft would be much more of a logistics challenge for supply than a smaller number of bigger craft (as in a carrier group).
I can't think of a scenario where the military invasion of Taiwan passes cost/benefit analysis for China, which is probably why it hasn't happened yet. The Chinese government strikes me as one of the more rational ones.