Current versions of OnSong only run on iOS/iPadOS 13 and higher, but you can continue to use OnSong 2020.5 on older devices. However, there appears to be an issue with the App Store and how older versions of iOS handle app updates where you may be prompted to install the latest update, but upon downloading the update, the device determines that the app cannot be installed. Tapping on the icon again repeats the process and never opens the app.
We are not sure why Apple has overlooked this, but it appears that the solution is to turn off automatic app updates. After that, tapping on the app should install the update but allow you to open OnSong afterwards.
Keep in mind that if you need to install OnSong on an older device that you cannot do so from the main App Store listing. However you can install OnSong from your Purchases onto older devices.