The Institute is destroyed on an extended timeline--rather than taking the in-game couple of weeks or days, it takes the SS almost a year of infiltration to successfully bring the Institute down from the inside.
During that year X6-88 has become...attached to the new Director, and has been made a little more open to questioning the truth of his existence. So when the Institute goes down, he stays.
That doesn't mean he's willing to put up with this nuisance of a man that insists on tagging along with SS, though. Even if his sense of humor is unusually...intriguing.
Deacon/X6, Enemies-To-Friends-To-Lovers Trope
During that year X6-88 has become...attached to the new Director, and has been made a little more open to questioning the truth of his existence. So when the Institute goes down, he stays.
That doesn't mean he's willing to put up with this nuisance of a man that insists on tagging along with SS, though. Even if his sense of humor is unusually...intriguing.