I was wondering if it would be possible for movie scripts that have been released for the public to be be put back in production. For example a few years ago when the potential for a new Friday the 13th movie was cancelled, the script was released to the public. So would it be possible for whoever gets the rights to the franchise to take that script and turn it into a movie? A similar situation happened in 2015 when a script for a new Halloween movie was released.
Your question is based on an inaccurate premise. Film studios are not in the habit of "releasing" unused scripts "to the public." There is no business reason for them to do so. In the example you cite, Paramount did not "release" Nick Antosca's script. The script was leaked. It was posted online by someone who had access to it, which amounts to illegal distribution of Paramount's intellectual property. Nevertheless, even though the script can now be found online, it remains Paramount's property.
If they chose, Paramount could indeed dust it off, spruce it up, and put it back into production. If Paramount were to give up the rights to the franchise, and those rights were picked up by another studio, those rights would be limited to the characters and trademarks of the franchise only. They would not include the use of Antosca's script. If the new studio was committed to that specific script, they would have to option that in addition to their purchase of the franchise rights.
It all really depends if the script has any proper copyrights and project seems like it is profitable enough to risk it even with the script leaked to the public. But if the project was cancelled but seems like it is profitable at a later date then there is nothing stopping them from restarting the project. It happens all the time in the movie business. But if the gap becomes too great from the previous movie than the new project typically does not do so well as the previous movies.