To add an app to a private channel on Teams, you must first add it to the Teams site. If the Team owner has restricted app installation to owners, the owner of the private channel will first need to request the app installation before it is successfully added.
Unfortunately, private channels have a site collection that is not connected to an Office 365 group. Because apps like Tasks have a tight connection to an Office 365 group, this means that Tasks is not available to run on a private channel.