Music app missing. App Store thinks it's installed

My iPhone crashed hard and I had to do a reset and restore from backup. Now the built-in Music app is missing. I had uninstalled it previously because I was trying to free up space (in the course of troubleshooting whatever the issue was before).

If I go to the App Store to reinstall it, it says it's installed. This is the *only* part of the OS that thinks this app is installed. If I click the Open button in the App Store it says:

Restrictions Enabled

Certain apps, features or services can't be seen or used when Restrictions are on. To use this app, turn Restrictions off.

I have turned off all Restrictions in the System Settings. I don't think it's actually a restriction setting. It's that the App Store is confused about whether it's installed or not. I have no idea how to get this app back. App Store won't install it because it's "already installed" (except it's not)