I’d like to point out that gen 3 synths can be effected by radiation. This is the main reason why the Institute hasn’t sent coursers into the Glowing Sea to end Virgil.
Also gen 3 synths are so human like that they can also bleed. This is because they are made of synthetic organic matter and not robotic like the gen 1 and 2 synths. So if the sole survivor were a synth, they’d be a gen 3.
But I don’t think that the sole survivor is a synth. While it is possible that the Institute could have taken the opportunity to replace them with a synth while he/she was in Vault 111, some time after they had taken Shaun, I don’t think that they did.
I’m not sure why VATS can be used before picking up the Pip-Boy though. The theory that the sole survivor is a synth comes from this fact and it is usually back up by a terminal entry that can be found in the Institute. The entry describes an improvement made to gen 3 synths (or maybe just the coursers, it’s not that specific) which sounds very similar to what VATS does. The theory mostly began due to DiMA’s questions to the sole survivor when they first meet during the Far Harbour DLC.
But there is some proof that the sole survivor is not a synth as well. What logical reason would the Institute have for replacing the sole survivor with a synth? And also, the sole survivor, if they were to be a synth, doesn’t seem to be aware of any such thing. The Institute makes sure that their infiltrator synths are aware that they are synths (eg McDonough, Art, Roger Warwick).