I think you're underestimating things here. These days we have two new milestones: a crewed mission to Mars, and a permanent moon base. The fervor to get there is of course nowhere near what it was like in the 50s and 60s, and I probably wouldn't even call this a new "space race", but the US doesn't want to look weak in matters like this, especially with China's general technological and economic rise over the past decades.
I actually think the US hasn't made such a big deal about Mars and the moon because they're afraid China will latch onto that and make it a big deal, and more likely "win". So instead we focus on LEO, ISS, Mars rovers, probes, telescopes, etc. And if China does eventually build a permanent moon base or get a crew to Mars before the US does, the US can claim "cool, good job, but we didn't think it was all that important, so we didn't pursue it as hard as they did".
(Note that I believe that Musk pushing hard for Mars is not the same as the US pushing hard for Mars. The US seems more outwardly excited about Musk's ability to cheaply ferry things to and from orbit than anything much farther.)