Dear Lexi Nickens,
Thank you for posting your query in our community.
Thank you for the details provided. SharePoint uses permissions to control user access on the site. If adding from SharePoint in the admin center still doesn't work, it may not be related to the team app, but rather to a problem with the permissions on the private site.
but because it's a private channel linked site, the permissions on the site are all set from the backend and users do not have permission to customize from the user end.
So in this case I'm afraid you need to check the permission with the backend team, if the permission is set correctly from the backend they will able to fix it.
If you are the Microsoft 365 admin of your organization, you can sign in to Microsoft 365 admin center with your Microsoft 365 admin account, and select Support > Help and support, enter your question and select Contact support. Otherwise, please refer to How do I find my Microsoft 365 admin?
Note: We apologize for redirecting you to related suport team because the moderators in this community are limited with the resource for further investigation and to get a quick resolution to your problem, we requested it.
If you think my suggestion was useful please consider clicking the below option.

if you need further assistance regarding how to contact our support team please feel free to visit our community.
Thank you for your time and positions we wish you have a lovely day.
Yours sincerely,
Sophia