Societies have practiced exposure forever, and it's not really about "survival of the fittest", it's about getting rid of people who aren't yet perceived as people (the massive death-rate in childhood had a lot of societal effects people are somewhat avoidant about discussing today) who might be a definite burden. It also gets rid of a lot of people who wouldn't be - many children were exposed simply because they were an extra mouth to feed. It's a sign of desperation more than anything else. The Spartans were unusual that they did it even when they didn't need to, but there's no evidence it particularly helped them.
The "deadly contests for their children" thing isn't really true though, and also didn't promote "survival of the fittest" (which is a pop-science misnomer anyway, the real phenomenon is "survival of the best-adapted"). As Sparta demonstrated very well, by the time the Romans had got there, the Spartan lifestyle had ensured Sparta was a pathetic wreck, not some sort of victory for eugenics - all the Spartans really managed to prove was that if you have warriors train pretty much all the time, i.e. a standing, professional army, brainwash/indoctrinate them into fanatics, and equip them with the best gear your society produces, they'll do really well in battle - they weren't the last society to show this, nor, I suspect, the first, just the first where it's well-recorded. And it had a cost, which they eventually paid.
And it's absolutely not "modern and enlightened" societies only, you're showing your ignorance about archaeology and history there. We've got bones from people hundreds of thousands of years ago where people with extremely serious injuries or deformities survived for decades, showing very clearly that even hunter-gatherers often helped injured people.