The chances of getting away with killing/maiming someone are higher for s/he who flees the scene

Well, only if the driver is intoxicated -- and I suspect that's the case in most of the hit and runs.

If the driver is not intoxicated, not provably texting, not racing, fleeing from the police, etc. -- they are almost certain not to face criminal (beyond a standard moving violation, though even that's fairly rare) charges from such a collision, so there's no real need for them to flee the scene.    They may get nailed in civil court, but that requires that they have assets to sue for, which often isn't the case.

(It would be nice if there was some legislation that increased the severity of normally minor traffic violations when they result in a serious injury or death.  Running a red light?  Class C misdemeanor.  Running a Red Light causing a serious injury?  Class B misdemeanor.  Running a Red Light causing death?  Class A misdemeanor.  A pipe dream, I know.  Though it would probably make hit and runs more likely as it would give even sober drivers another reason to run -- as an even sober driver could be sent to jail.)

Amnesty (for DWI or whatever else) for drivers who don't hit and run could fix the problem, but it goes way too far.  I mention it as a possible solution, but I think it's even worse than the problem.

I think a more fundamental issue is that legally it's hard to prove "beyond a reasonable doubt" that somebody knowingly fled the scene unless they're caught by the police almost immediately affer (which usually means finding that they were drunk) or they unwisely confess to the crime.

Once they've had a chance to sober and lawyer up, even if the police find the damaged car and match it up to the collision (likely!) ... they have a hard time proving that the owner was the one driving it.  The DA could prosecute such cases more aggressively, but they'd likely lose most of the time, as the defense would (properly) bring up that there's no evidence about who the driver was.  They generally lose any civil cases -- the standard of proof is lower, the owner of the car is generally at least somewhat liable for it's misuse anyways (and the car is probably insured anyways, whomever the driver is) -- but they don't go to jail.

Short of constant, total surveilance or fundamental ("beyond a reasonable doubt" and "right to remain silent" are mighty fundamental) changes to our Constitution, I'm not sure how this can be really fixed.  I don't think the DA and police can really fix this one.  The legislature can fix things to a degree (for starters, make the penalties for leaving the scene at least as serious as a DWI -- you can't really presume that one is DWI, but you can make the penalties as bad as a DWI), but I don't think they can really properly fix it.

I know that some people has been calling attention to the problem for ages, but they are voices claming in the desert.
Sad.

Seems like it's almost part of the culture now to try to flee if you hit somebody with your vehicle.  It's hard not to be a misanthrope when I have to remember that about half the people out there, if they hit me, are quite willing to leave me for dead.