Hi,
Thank you for posting in our community.
Since you mentioned ‘’Our goal is to make the site accessible to anyone in our organization, but have restricted access to our documents.”, I would suggest you to follow these steps to grant permission to your user again.
1. On the SharePoint site Home page, click the gear.
2. Click Site Permissions, and then Advanced Site Permissions.
3. Click Grant Permissions button to open the Share dialog and then Invite people.
4. Type Everyone except external users and then click Show Options, select the permission level to Visitors group.
5. Click Share and then all users in your organization will receive the invite email.
6. Ask them to access the site by this link.
Moreover, if your idea of “Restricted Mode” is to restricted access to your documents, I would suggest you set specific Permissions levels on the related Document Library page.
For example, to set unique permission for a Document Library,
1. Click the gear then Library settings> Permissions for this document library.
2. Click the Stop Inheriting Permissions button.
3. Select all existing users and click the Remove User Permissions button
4. And now you could grant permission to specific users by clicking the Grant Permissions button.
If you need further help, feel free to contact us.
Best Regards,
May