Ok I may have missed something but how can Father be Shaun? Here's what I got from the story:

Kellog stole Shaun from the cryo chamber.
You find Kellog (who looks the same as when he stole Shaun) and then try to find the institute
You then eventually get to the institute where "Father" is apparently Shaun and it's actually 60 years later

How the hell is it 60 years later if Kellog kidnapped Shaun as a baby?? Kellog sure does not look 60 years older. So either I missed something like some info about Kellog and his implants, or there's a serious plot error here.

We have a section for spoilers, might want to post there

Kellog had a lot of mechanical parts in him, and had the assistance of what is likely the most technological group seen in the wasteland, even above the Enclave and BoS combined, it is more then believable tha he is alive that long.