A public space race of billionaires is exactly the kind of thing we should all be hoping for. They can leverage their relatively small funding to push much larger government funding in productive directions. They're better than most government administrators and don't have as many constraints, so they can take more risks and pay experts what they're worth on the open market.
The public should be calling for more of this kind of thing across all major areas of civilization. Let's get a public race among billionaires for curing cancer, HIV, and poverty.
If it feeds their egos, all the better. Almost all progress is connected to selfish goals, even among the greatest and most "pure" scientists. They want to win a Nobel, for the money and the recognition. Or they're doing science because it's just extremely pleasurable for them, which is not a selfless reason at all. The sooner people realize that base motivations aren't a sin the better we'll all be. And in reality, motivations are always a mix of (mostly) selfish and selfless reasons.
Motivations aren't what matters. What matters is making progress that can (eventually!) benefit everyone.