Will I be absolved of responsibility before Allah in this way if transgressions occur, despite my diligence in preventing access to websites that contradict ethics, Sharia, and public order, given my inability to monitor conversations?

The "pass card" limits access to bad websites, but it does not absolve its issuer before Allah if he cannot monitor what the center's users say and write, especially if they are immoral individuals who do not hesitate to do anything. This is because the lack of awareness of Allah's greatness and His watchful eye, and the lack of fear of His punishment, pushes a person not to adhere to what he has agreed upon.