I have recently encountered similar problems with apps published to a private audience and have made several attempts at contacting Microsoft Support in hopes that they can find a solution, but to no avail.
To frame my issue specifically, apps published to a private audience suddenly stopped appearing when searched for in the newest version of the Microsoft Store, and the direct link to the app also ceased functioning for some users.
Similar to your issue, what causes the broken link appears to have something to do with an organizational email address being used for the Live ID, but that has not always been the case from my experience. What I found instead was that the direct link appearing in the Microsoft Partner Dashboard is no longer valid depending on what version of the MS Store app is in use.
Workaround:
- Make sure that all updates have been applied to the system (Windows 10 or Xbox One)
- Manually check for updates to the Microsoft Store app and apply those if available.
- Retrieve a direct link to the app from the Microsoft Partner Dashboard. Application Overview > Product Management > Product Identity > URL if your app is only visible to certain people
- Verify the link because it may not be correct. For example, it gives me a link in the following format: https://www.microsoft.com/store/r/{PRODUCTID} When it should be in this format: https://www.microsoft.com/en-us/p/{APPNAME}/{PRODUCTID}
- On a PC (Mac is fine if you intend to install the app elsewhere), use the link to navigate directly to the app in the Microsoft Store website. If it has not already been acquired, then a get button should be there. Clicking it initiates a purchase flow. Even if the app is free, make sure to follow through.
- If the application is to be installed on Windows, then you should be able to install it from here.
- If attempting to install on an Xbox One then you will need to proceed to the next steps. Remember that you need to update the system and the Microsoft Store for this workaround to be successful.
- Open Microsoft Edge on the Xbox One and type in the direct link. If all is well, then the product page for the app should appear and clicking the button will redirect to the Microsoft Store app on the Xbox One.
- Install the app.