I think the intended reading is (in my own rewording) "whenever the Blazesteel Archon is flanking a foe, each time it would be able to choose to make attacks add +1 to the number of attacks made. For example: on its turn it, due to its base attack bonus, if attacking with a weapon it normally makes 2 attacks, while flanking it makes 3. If on the enemies turn, it is still flanking and its foe (that its flanking) causes the Archon to make an attack of opportunity, it would get 2 attacks, 1 as normal +1 for its flanking bonus." This means its potentially more than just "an extra attack per round" as you said, but not nearly infinite, which yeah the poor original wording can be abused to imply.