You might not have to worry about apps that can't use the 120Hz refresh rate of the iPhone 13 Pro family. As iMore notes, Apple has posted a developer article revealing that iPhone apps will need to 'unlock' 120Hz support by adding a key to a .plist file. There's also a bug limiting the speed for some Core Animation-based apps, although Apple said a fix was coming in a future iOS 15 update.
The iPad Pro doesn't require code updates to make full use of its 120Hz display, although that may be due to the tablet's much larger battery. Smartphones are much more likely to take a serious battery life hit from high-refresh screens.
The details suggest a repeat of what happened when Apple introduced the iPhone 6, iPhone X and other phones with significant leaps in display tech — it took a while for Apple and developers to make full use of the new screens. Your iPhone 13 Pro's 120Hz panel should reach its full potential, but there may be a few weeks or months where it goes underused.