Who said that Nagato "unlocked" his rennin, id like to see the manga scan of this.
Also, Nagato needed to learn how to control the powers his eyes gave him, thats the real and only unlock that nagato could do.
Rennin can be turned off, at least if its a natural progression from EMS like madara's was (by natural i mean, doing what madara did to get his ems), notice, when madara was fighting the kage and naruto along with the whole ninja alliance he did not have his rennin active the whole time. So yes these eye eyes when acheived naturally can turn off, if they are not achieved naturally, such as kakashi, Ao, Obito (rennin), and Nagato, then it will not be able to be deactivated.
Also the situation that you described would not lead to a plot hole. A plot hole is a situation that makes no sense and no real solution can be made with information provided, such as, out of no where with no explanation naruto obtaining byakugan, if naruto meets with Hamura and then appears with Byakugan then there is no plot hole because a logical reason as to how naruto got the byakugan would be that he received some powerup from hamura that unlocked the byakugan in him.