I was thinking more along the lines of sentience developing on its own. It would be stupid for the Institute itself developing it. I'm thinking along the lines of Robocop and how certain memories came back but he never really found his former self.
I suppose it could feasibly be integrated as part of the Railroad's questline to find a cure and it could develop into something interesting if the Institute catches wind of it (or decides to tell you about the defect, as I'm sure they'd already know); after all, the human brain is a fickle thing.
My main issue with the Institute is that they never seem to act in their best interest; they just seem to commit random acts of violence for no reason. Even fearmongering isn't a potential excuse because there's no possible reason why they'd need the people of the Commonwealth to be scared of them.
Or, to put it another way, they're simply too overpowered.
They're so technologically advanced in so many areas it makes you wonder why they'd need to interact with the surface at all. The only possible reason I can think of is sample collecting and there're a million different ways for them to do that without revealing themselves.
Here's an example; why didn't they just bomb the Prydwen with their teleporter? You can comfortably travel to the Institute from anywhere, so it follows that they can do the same. I do seem to recall the Brotherhood having some sort of anti-teleportation thing going on (miraculously, given how when you found Danse they didn't even know the Institute was a thing) but why can't they just send a synth agent to disable it? Or just unleash a ton of robots and super mutants? Or detonate a bomb in close enough proximity to the field projectors to destroy them?
Assuming that the Railroad was piss-easy to find for gameplay reasons and is in actuality pretty hard to uncover, why not have synth agents claiming to be refugees in order to find them? Why not send a Courser (allegedly unstoppable badasses) to follow this "freedom trail"?
Also, why was finding Virgil such an issue? Nick himself says that synths are immune to radiation, so what's the problem? It can't be the wildlife because, again, the Institute can just teleport wherever they like; even if they don't know where Virgil is they can just keep looking and instantly escape and return to the same spot later if things go south.
In essence, why would the Institute care about what's going on topside and why are they having such a hard time achieving their goals?
TL;DR Institute is dumb