Currently, Treasury Board is responsible for administering access to information, but each institution must have an operations unit that is in charge of implementing procedures to respond to access requests. I don't know if an independent institution could get access to the institutions' documents. It would be difficult.
To go back to the beginning of your question, I would say that the Access to Information Act was put in place to provide access to information, with certain limited exceptions. Since 1983, people have seen the act instead as a way to prevent access to information. Rather than asking what information should be given out, we ask what information should be protected or exempted. The application of the act did not allow it to achieve its ultimate goal.
Could having a central unit in the government allow for better administration of access to information? It's possible. Some provinces, such as Alberta, have a central unit, but they still have to check with the ministers responsible for the institutions. Ultimately, the decision rests with the head of the institution. I'm not sure it would change things. What needs to change is the culture within government. We need to adopt a culture of transparency that would make people realize that it's a good thing to share information and to ensure that political parties have the information they need to have a public debate. It's good for democracy. However, right now, I think people are afraid. They hide behind the Access to Information Act, and that's why there are limits and why it's not a perfect process.