For decades, Americans have been propagandized into the position that they alone are exceptional. Thus, anyone that challenges that belief becomes the enemy. It's gotten so bad, that "China might get there first" is the only way to get American politicians to actually stick to a target for more than 4 years.

Depends on your point of view. But I imagine some people in the US would not be happy to lose that race. The reason it was a race in the sixties is because they definitely didn't want the Russians to get there first.

China isn't looking at it just for bragging rights, but as a step towards the first moon base. Some see it as a race for the frontier and territorial claims.