No, that’s definitely not it. Even if you stop open registration, those subscribers will remain as they are. Closing a means of accessing your site doesn’t do anything to the status of those already on it.
So let’s take it in steps.
First, if you want to stop the free-loaders accessing your content, you can simply protect all of it at level 1. Those subscribers are at level 0, so they won’t be able to get to it.
Then you can add a Billing Modification form to the Login Welcome Page, enclosed within shortcode conditionals so that only subscribers see it. This will enable those members to upgrade to a paying level.
As for not being able to prevent Open Registration, there are two primary possibilities. One is that your installation of s2Member is corrupt. If s2Member is working properly otherwise, though, this is unlikely. It typically happens only if you installed s2Member in a non-standard manner anyway.
More likely is that you have a theme or plugin conflict. Try turning off all your other plugins and changing to the 2017 theme. If you can close registration then, you will know the issue is a conflict. Then turn things back on one by one and test each time until the problem recurs. There’s your culprit.