Not to put too fine a point on it, but the relevant criterion here isn't what's best in some abstract sense, it's what governments actually implement, and to a lesser extent what our intuitions about punitive legitimacy say. Cultures vary, and so does rhetoric, but I've never heard of a justice system that wasn't functionally motivated at least in part by retributive sentiment.

where the system is designed

You lost me at "designed".

I think this claim is either tautological or false. The tautological sense is that almost any goal, including abstract justice or state-sponsored retribution, can be viewed as ultimately contributing to a healthy society, thus protecting people. I would be surprised to find many advocates for any criminal-justice policy admit that the long-run consequences of their policy are bad.

If you take a narrow view of protection, then the claim is false. Advocates of some current or proposed US criminal justice policy often make arguments that leave the connection to "protection" fairly vague. To pick one example that's been in the news lately, let's look at capital punishment. I don't normally hear death penalty supporters talk about protecting society from the person sentenced to execution. They are usually interested in the abstract symbolism or "morality play" aspect of a bad person being punished for their wrongdoing. And you might think the social consequences of this spectacle are worthwhile, but it doesn't see to be about "protection", narrowly understood.