Old app didn't transfer to new iPhone -- workaround?

There is not. Not a legitimate way. If a high school kid has a method it would be using an unauthorized alteration of the iOS…which you will not and can not get help with from here.

Check to see if the app is indeed no longer available in the App Store to redownload. If it is you will see the cloud to redownload it. If it isn’t you won’t find the all at all in the App Store.

View your purchase history for the App Store and other Apple media services - Apple Support

View, redownload, and manage apps in the App Store on Mac - Apple Support

Because it’s not available to download again you won’t be able to get it on the new device. You can double check to see if it is indeed no longer available in the App Store. Otherwise, you won’t be able to get it again. It continued to work on the old device because it was already downloaded but if you had deleted it and wanted to redownload it you wouldn’t have been able to.

The best of my knowledge that will not do anything to put the app on the new device. App data and settings are saved, not the actual app. If it’s not available to redownload it doesn’t matter if you restore a full backup from a Mac or computer….the app itself will not be added to the new phone.

You’ll need to wipe everything on the new iPhone and effectively start from scratch. Backup the old iPhone to a Mac and then install the backup on the Mac to your new iPhone. Apple has several support articles i can link to if you want to go that route.

You can also just look in the App Store app under Account, Apps, My Apps and then scroll down or put the name of the app in the search field. If it’s still available to download the cloud will be there. If not it will not show anything (cloud will be greyed out) and if you click on it you will just get a message like “Cannot Connect This page could not be loaded”.

Um.. no. Apps don’t back up to iTunes. That won’t work.