Great answer, and I think there are much broader reasons too:
International order is fundamentally anarchy. There is no power - no world government - that can enforce order (the United Nations is just an association, a place where sovereign countries can meet and make arrangements with each other). If the order breaks down, if 'might makes right', then international wars start; wars are bad or catastrophic for everyone. WWII was so destructive that the victors worried that the world could not survive another one (one reason they started the UN and what became the EU) - and that was 75 years ago. The 'US-led rules-based international order' strives maintain an order based on rules (laws), not on might.
(Yes, this is all very simplified; motives and actions are not so simple.)
A simple example is when Iraq invaded Kuawait in the early 1990s. They had the might, so they took it. The US, openly stating they would maintain international order, drove them out of Kuwait. To reinforce the point, they did it according to international law, and with a coalition of international partners.
China seizing the South China Sea is a serious threat to that order - what will China seize next? If the only way to stop them is to fight a war - if there is no rules-based way to do it - the world and the people in it are in big trouble. So the US and allies (which is not only western, but democracies all over including Japan, India, etc.) work to undermine that claim.