Does it say anywhere in FM 7-22 that Physical Fitness has to be organized?

It states that throughout the book, starting on page 1-9:
The objective of PRT is to prepare Soldiers to meet the physical demands related to mission and C- or D-METL. This occurs through an organized schedule of prescribed PRT drills and activities.

However, Commanders can authorize individual PRT:
page 4-4:
Commanders that authorize the use of individual training programs for their Soldiers should follow the same training management principles outlined in the previous paragraphs

It’s up the the Commander. The Unit’s fitness program is his lane brother.