Google search results do not include the canonical (non-AMP) URL in the search results, which makes is impossible for Overamped to know the correct URL to redirect to. Without access to “All Websites” Overamped will make a best guess to “de-AMP” the URL, but this will not always work.
This is because not all website use URLs that can be “de-AMP” using a single algorithm. For example for an abc.net.au article Google provides the URL https://amp.abc.net.au/article/100303036 but removing the amp. subdomain will yield https://abc.net.au/article/100303036, a URL that cannot be found (also known as a 404 error).
When access to “All Websites” is provided Overamped will skip the Google AMP cache but will still load the AMP version of the search result. As soon as the AMP version starts to load it will read the URL of the real (canonical) version of webpage and redirect to it.