It is true that iTunes and Finder were written in Carbon, but it is also true that Apple's original plan was for Yellow Box (the OpenStep API that later became Carbon) to be the main API for developing programs under Rhapsody, which was essentially NeXT with Mac OS 8 theming and UI guidelines. Rumor has it that Adobe and Microsoft balked at the prospect of having to port their applications to Yellow Box (there was no guarantee in 1997 that Apple's purchase of NeXT would pay off), which led to Apple announcing its Mac OS X strategy during WWDC 1998, which included Carbon as a first-class API for the development of Mac OS X applications alongside Cocoa (Yellow Box).

Regarding iTunes, one detail about Carbon that is sometimes forgotten is Carbon originally ran on Mac OS 8 and 9 to facilitate the transition to Mac OS X (https://en.wikipedia.org/wiki/Carbon_(API) ; to quote, "Carbon was introduced in incomplete form in 2000, as a shared library backward-compatible with 1997's Mac OS 8.1. This version allowed developers to port their code to Carbon without losing the ability for those programs to run on existing Mac OS machines.") Given that Mac OS 9 was still current when the first version of iTunes was released, it made sense for iTunes to be developed as a Carbon application (since Carbon applications can run on Mac OS 8 and 9 as well as Mac OS X) rather than as a Cocoa application (which can only run on Mac OS X).