@midnight42 That is a very good question.
TL;DR: I don’t know.
Recently, a majority of the older games were removed from the iOS App Store. If you had purchased/downloaded games previously, the respective games were available for download in your Purchases section.
Initially, Rovio stated the removal was temporary, and, a short time later, the games were added back to the Store. (see following thread: Is AB Seasons off the App Store?)
Unfortunately, a few days ago, the games disappeared again. However, this time, my Purchases section lists all my previously purchased/downloaded Rovio games except the original Angry Birds. Both the iPad and the iPhone versions are absent.
I reviewed several articles on installing older iOS apps devices, but with Apple’s change to iOS synching, it is becoming more difficult to find a workable and/or acceptable method to do so (i.e., installing backups, etc.)
After the previous game removal, I downloaded every Rovio game to all my iOS devices just to be safe. So, I don’t have any devices to test alternative methods, but I did find copies of older Angry Birds app versions in my Backups. I may give it a go to see if there is a way for those who do have old versions available.
Currently, I believe the best option is to convince Rovio to add the game back to the Purchases section. Hopefully, other more tech savvy Nesters will have better advice. In the interim, I will do my part and send Rovio a customer service request to add the original game back the Store or at least to Purchases. If they receive enough requests, maybe they will add it back.