That's why it doesn't make sense!!! Because the subject of the sentenct is "club", you can't read the sentence to imply the cumulative action of more than 1 club without it being stated somewhere in the sentence. It's not. Simply making clubs plural causes a technical problem within the rules that we know isn't possible. Therefore we know the subject is singular. Then we should also understand that the rest of the sentence applies to the subject and not the object of the verb (players). In other words, club designates (player for a third time). The action is in parenthesis. If the action is carried out by the club for a third time.....
Our reading comprehension and use of shorthand has gotten to the point that we understand more than what is written. That sentence as it is written implies exactly what it means. One must read more into it to make it sound as if "a player franchised for the third time ever" means the same as "any club that designates a player as the franchise player for the third time."