Punitive damages going to the government wouldn't work. Why would a plaintiff and their lawyer spend their own money to try to prove punitive damages if they weren't going to benefit? Asking them to work for the government for free wouldn't be tolerated in any other context--would you agree to work for nothing on behalf of the government? I'm not anti-government by any means, but I sure wouldn't.
Punitive damages seekers are often referred to as 'private attorneys general"--they get punitive damages, and as a result, we are all better off because the defendant has been deterred from tortious misconduct in the future.
Now, I think a better question is why the first plaintiff to sue for punitive damages will get them, but plaintiff number two does not, since the defendant has already been assessed them. I have always thought that there should be a punitive damages pool, overseen by a special master, to divide punitive damages among similarly situated plaintiffs rather than reward a race to the courthouse. No jurisdiction has chosen to do this, though; probably because administrative costs would be too high. It would be fairer, though...