The Institute's Director at the the time of Shaun's kidnapping was a woman. In the Director's Recording 108 holotape she rants about how the accident with the prototype in Diamond City threatens her legacy and decades of work as Director. The Broken Mask incident took place in 2229, two years after Shaun was taken from Vault 111.
You left out the rest of the post:
"It wasn't my idea to settle down with kid in the middle of Diamond City. I thought it was a terrible idea, actually. But it was one of the old man's pet projects, so here we were. Me and the kid like a happy little family. I ended up liking it a reminder of what things might have been if things had turned out differently. But there's no going back. I knew it was just temporary, and it would be back to normal business before too long. This whole setup was part of some elaborate plan of the old man's. Seems obvious now we were bait for our old friend from the Vault. The timing couldn't have been an accident. That's not how the old man works. I wonder if he outsmarted me in the end. Another loose end tied up."
This is all dialogue from the same quest. There is no question that "the old man" is meant to refer to "Father" Shaun here, right?
Back to the quotation in question:
"I never knew why we didn't just refreeze the rest of them, but we had our orders. I guess the old man didn't want so many loose ends. Too bad he left alive the one person he shouldn't have."
Given the entire context, this implies that Shaun decided it wasn't worth the trouble to continue keeping them on ice when all he was interested in was keeping his parent alive to experiment on. Kellogg is reflecting on the vault dwellers who, like him, are "loose ends" that "the old man" decided to kill. So yeah, I think sixty years after the kidnapping Shaun decided to order his parent released while everyone else in the vault got killed. The other explanation is that there was more than one "the old man." I've already given my thoughts as to why that seems less plausible. Naturally, ymmv.
Really, the whole thing seems ridiculously contrived so that the sole survivor is the sole survivor. There's no justifiable reason for Shaun or anyone else in the Institute to needlessly murder them all when there are better alternatives like simply releasing them. But taking the dialogue we're presented with in the quest, Shaun seems to be the one left holding the bag, and I can see how it fits his character's views to make that decision.