The concept is there. Jesus was baptized publicly as a statement of the beginning of His ministry. In Acts 2, three thousand believed and were baptized, hard to not be public. Acts 8, the eunuch was baptized by the road. Paul, when his name was still Saul, was baptized right after gaining his sight. So there were most likely others there to take care of him. Acts 10, Peter baptized with others present.
Yes, the Bible does not say explicitly to baptize in public, but why would you want to hide your faith? The baptism is the representation of you dying, like Jesus, buried when you are in under the water, like Jesus, and raised with new life, like Jesus was raised. It is a symbolic profession of the faith you have received.
Not sure why you would argue/question those points? If you have scripture to the contrary I would love to see them so I could learn more about baptisms.