So, here is my theory - totally inferred from just a few statements by Father and fits my head canon, so I'm going with it.
By the end of the game, Father has come around to realizing that synths are sentient and that he and the Institute have done a terrible wrong by keeping them as essentially slaves.
My reason for this conclusion is synth Shaun.
When you listen to the holotape that synth Shaun gives you from Father - Father speaks of Shaun, not as a piece of machinery or a bit of cool new tech, but he speaks of synth Shaun as a person. He says:
Nuclear Option: If you are hearing this, then whatever conflicts you and I have endured are over. I have no reason to believe you'll honor the request I'm about to make, but I feel compelled to try anyway. This synth, this... boy. He deserves more. He has been re-programmed to believe he is your son. It is my hope that you will take him with you. I would ask only that you give him a chance. A chance to be a part of whatever future awaits the Commonwealth.
Nuclear Family: If you are hearing this, than the time has come. I am gone. I can no longer look after young Shaun. I hoped that you might be willing. He has been reprogrammed to believe he is your son. I hope that was not too presumptuous on my part. Both he and you deserved a chance to... To be a family. Please, take care of him.
The fact that Father speaks of Shaun as "deserving more" indicates an awareness that he is something more than just a toaster or even an AI like a Mr. Handy - machines do not deserve anything - we have no moral debt to them, but Shaun, in Father's view deserves a chance to "live", a he deserves a future. This is a big step away from Father's previous rhetoric that synths are merely property. Father had previously admitted that synths AI were "approaching sentience" so now perhaps Father realizes that Shaun is something more than just a machine. Something more than a cyborg facsimile of son you lost - but an actual boy (or something very close to it) in need of a parent.
So, in addition to entrusting Shaun to your care, Father also gives you control of the Institute knowing (depending on your gameplay choices) that you may very well take the Institute in a new direction regarding how synths are viewed.
Father, at one point, tells you that here at the end of his life, he has regrets and that is one reason that you were thawed out. Perhaps one regret is what has happened to the synths under his leadership and he hopes you can take the Institute in a new direction (if you side with them - otherwise, he just want you to give Shaun a chance at life).
So, I think Father's views on the synths are evolving and complicated and the way he speaks of synth Shaun indicates that he realizes that they can be something more than just machines and so he entrusts Shaun and the Institute to you to care for.
This theory works for me. Perhaps it is just head canon or wishful thinking, but I think that a case can be made for Father wanting you to move the Institute in a new direction recognizing 3rd gen synths as something more than machines, but as persons that deserve more than what they have so far been given by the Institute.
Of course, there is no gameplay mechanic to support this - the game does not allow for us to sit with the department heads and shape new policy for the Institute, but from a narrative point of view this supports my pro-Institute but also pro synth role-play.