I get that impression as well. One of the reasons why I don't like the Institute.
As I said before if they wanted to stop the Synths it would be incredibly easy to do so.
Hell they could take out the Rail Road so easily. Step one make a Synth with a very sad story, place a nuclear bomb in it. Step two: have it find the Rail Road. Step 3: Boom!
Hell just do what the Terminators did and send a bunch of Coursers out as normal Synths. Once they find the Institute, kill them all.