South Korea and especially Japan are much closer to being able to build nukes, if they so desire. I personally heard the former US defense secretary, Robert Gates, stating that he thinks Japan can get nukes in a matter of weeks.
So, if China takes over Taiwan, there won't be anything in the world to stop Japan from getting nukes, and South Korea may join the party too.
[1] https://en.wikipedia.org/wiki/Timeline_of_the_Republic_of_Ch...
It is also ridiculous to say the U.S. conducting the Freedom of navigation, because the U.S. has not signed nor ratified the convention. The U.S. only use its name to enlarge its militarily pressure.
EDIT: forgot to mention not the Taiwan Straits nor the South China Sea is international waters, they are Chinese territorial waters.