I want to keep the site private until it is ready to go public; even if I make the Notion side of the site private, the site content will remain.

We apologize, but Wraptas does not currently have the ability to make a site private.
In addition, Wraptas has a cache function to prevent the site from being deactivated in the event of an error or mistake on the Notion side, so even if the former Notion is deactivated, the previous site will remain.
We are planning to provide a forced update function from the administration screen in the future, but since we do not have such a function yet, please create an empty page in Notion and set it up as a temporary measure.
If this is too much trouble, you can set an empty page URL here.