While it is true that the Institute may not ethically be the best faction and that their leaders are flawed, a change in leadership is a more sensible and effective solution to this problem than blowing up the Institute with a nuclear explosion and causing severe damage to a highly populated area of the Commonwealth in the process.
The Sole Survivor's connection with Father in addition to the fact that there are already scientists within the Institute who would favor the Institute helping the people of the Commonwealth (e.g. Dr. Li and Dr. Karlin), means that turning the Institute around is not really that far fetched.
And then there is the argument about synths. Synths are androids, or biological robots; robots designed to look and act like humans. Just because the Institute has succeeded in making them do just that, that doesn't mean that they are actually real people. They are advanced artificial intelligence; near-perfect simulations of human intelligence and as such are expected to behave very similarly to real humans. However, a near-perfect simulation does not equate to the real thing. Just like how a flight simulator is not a real plane, a synth is not a real person, and for the exact same reasons. No artificial creation of mankind will ever be granted personhood.
And even if you believe that synths are people, the Railroad is still wrong in what they are doing.
Dr. Binet is the only scientist in the Institute who believes that synths are people and even he is against the Railroad. The Railroad is a complete and utter joke. They have nothing that is worthy of respect or endorsement. They are incompetent and their ideals are cringe-inducing, at least the other factions are not a combination of both. I can at least respect the Brotherhood of Steel for their military prowess and for defeating the Enclave in the Capital Wasteland. Speaking of the Brotherhood of Steel, their argument against synths is also flawed. Synths are not doomsday weapons, a Brotherhood Paladin is roughly an even match for an Institute Courser. A 3rd generation synth is no more dangerous than a military trained human.