In response to the OP...
Valentine is actually the character who changed my mind. Prior to meeting him, I thought synths were abominations. I mean, think about what you know about the Institute early game, the stories you hear about synths in Diamond City, and other various bits of information. For someone like myself who RPs, it was very easy to associate synths as an evil product of an organization that killed my wife and stole my child, effectively ruining my life. Valentine changed that perspective.
From the outset, it was unsettling to see the detective who was going to help me was a synth, but I thought this would just be a path through the main story that I had to take, so I didn't have any expectations. Valentine proved himself to be an incredible guy though, helping you track down Kellogg and sticking with you after. He has a good moral sense and I like the observations and knowledge he has about the places we would visit. What really sold me though was his own self-awareness, humility, and goodness. Valentine recognizes that people have a reason to mistrust him, he doesn't place any expectations on them, and did a lot of dirty work to earn the communities trust before he was made into a detective. He is very grateful that the citizens of the Commonwealth have been so good to him, because he knows it very easily could have gone down differently, and he fondly remembers the first child who had the courage to talk to him. Valentine is a beautiful person in other words.
As the OP says, he doesn't remember being from the Institute, and he is definitely a prototype of some sort, but that doesn't really make him less of a synth right? The whole argument against synths is that they are either abominations or sleeper cells for an insidious movement, but Valentine flies in the face of all that and really showcases how synths have all the same potentials and pitfalls as any other human being. I joined the Railroad because the other organizations wanted to subjugate or destroy all synths. I don't see any reason to do so after meeting Valentine. I don't know that I feel as strongly as the Railroad does about the need to liberate synths, but if you think about their plight (conscious beings subject to the will of their masters for fear for their own lives) it does make me feel motivated to help at least. Like Desdemona says when you meet her, the Railroad is the synths only friend in the Commonwealth. I feel like combined with the Minutemen, the Railroad is the best choice for which faction to support.