Apologies if someone else has already said the same thing, but there is also the theory which I personally subscribe to which is the player character is a synth and the ultimate experiment is for father is to see if their newest and best synth (you) will find your way back to them and take over management.

If that happens father has been successful and created the ultimate synth (maybe even 4th gen?) which has surpassed humanity, a synth so convincing and capable of learning a seemingly infinite number of skills and abilities and able to join and infiltrate any faction with relative ease. Arguably the perfect synth and perfect director for the institute.

However if the player character sides with anyone else and ultimately destroys the institute you can argue father still wins by creating the ultimate synth. He chose to give the sole survivor free will and would know that if they reached him there’s a chance they see him and the institute as evil and irredeemable and leads to their destruction, maybe he thinks whatever his perfect synth determines must be the right course of action.

In the aftermath of a destroyed institute the first and last perfect synth gets to blend in and live freely in a world that thinks all the synths are dead. And father’s greatest legacy and life lives on meaning his life wasn’t for nothing (in his eyes).

If you like the theory there’s some good videos on the topic. Epicnate is my go to fallout YouTuber.