I imagine that lack of extradition treaties is somewhat of a hindrance for establishing visa-free travel between countries, on the argument that if (potential) criminals can enter freely (and then do some crime) one should be at least able to get them extradited thereafter.
The purpose of visas and of visa exemptions has little to do with criminal justice or with the purpose of extradition treaties.
To the extent that a country uses visas to keep undesirable foreigners out, and to the extent that some foreigners are undesirable because they will commit crimes while they're in the country, consider that the existence of an extradition treaty with the foreigner's country of citizenship doesn't make it easier to get the foreigner out of the country after conviction, because that's not extradition; it's deportation. No treaty is needed.
The purpose of an extradition treaty is to get someone who is accused of committing a crime in your country into your country so you can prosecute them. (You can deport them afterward, if they're not a citizen of your country.) The only advantage of an extradition treaty is that you are more likely to be able to bring criminals to justice who have visited your country and committed a crime there and then left the country before they could be arrested.