They wouldn't claim the Philippines without a war.

But the Chinese government is trying to claim territorial waters, and their aggression toward all their neighbors is exactly why the Philippines have decided to work with the US and open new bases.

They may not claim it, but they are definitely the aggressor in the territory, and most of their neighbors do not view the Chinese government as 'the good guys'; that's not to say they view the US as 'the good guys' either, simply as the one that isn't actively aggressive and could assist in their defense.