Whatever the reason why people didn't think Jesus could be the Messiah, after Peter says that he is the Messiah, the first thing Jesus tells his disciples is not to tell anyone else that he is the Messiah. I suspect there were a couple of reasons for this. The first reason was that people would then gather to him in the hope that he would start a rebellion to drive the Romans out of Israel. And that was not what Jesus was about. In fact, from this point on Jesus repeatedly tells his disciples that he must go to Jerusalem, where the priests will hand him over to the Romans, who will execute him. Whatever his goal was as the Messiah, it was not a political/military goal. He was not trying to free his fellow Jewish people from Roman rule. He was trying to free them and all of us from something else - sin.
I think the second reason Jesus didn't want people saying that he was the Messiah is that it would have made him an instant target for the Roman rulers. He knew that eventually they would kill him, but he wanted it done on his terms, at a certain place and time - in Jerusalem, at Passover.