Not true. Many professional musicians (like myself) routinely have to sing or play at various pitch levels: A=390Hz, A=415, A=420, A=430, A=435 or so, A=440, a bit sharp of 440 (European orchestras), etc. Many of those musicians have perfect pitch to varying degrees.
Having perfect pitch does not make it impossible, or even difficult, to "enjoy" music at varying pitches, etc.
I think it depends on the musician. A couple of friends of mine struggle against their absolute pitch in historical pitch contexts (especially a notorious 19th century organ tuned to A450) and have to think in microtones, others just dial in and deal with it.
Having perfect pitch does not make it impossible, or even difficult, to "enjoy" music at varying pitches, etc.