I believe that's the inevitable outcome.

The Chinese will build a moon base, as a sign from the Chinese government to the Chinese people that China is capable of cutting-edge engineering and science (notably a demonstration to their own citizens - when was the last time you heard about the Chinese space stations outside China?).

America seems a bit shaky in their determination to actually build a moon base, though having Jared Isaacman as administrator gives hope. But regardless of whether America is currently on track, a successful Chinese moon base won't stay without answer