Except it's p. 127 says 'once on each of your turns'. It's not once per round or once on your initiative.
I am not gonna try and argue it, but a reaction is not your turn. That’s how the game works.
I think you got something wrong.
EDIT: Couldn't stop myself... This is from the new book:
Reactions (page 15)
Certain special abilities, spells, and situations allow you to take a special action called a Reaction. A Reaction is an instant response to a trigger of some kind, which can occur on your turn
or on someone else's. The Opportunity Attack, described later in this chapter, is the most common type of Reaction.
When you take a Reaction, you can't take another one until the start of your next turn. If the reaction interrupts another creature's turn, that creature can continue its turn right after the Reaction.
In terms of timing, a Reaction takes place immediately after its trigger unless the Reaction's description says otherwise.
Nowhere does it say that it becomes your turn.