Have you noticed that few of the records show a source at all?:-)
Yes, most churches kept records of baptism, marriages, and funerals/burials.However, some churches are very secretive about disclosing private information, although that varies a lot with the current record keeper.
New England Historic & Genealogical Register has many early church records, and Mayflower Descendant magazine also has many, not just records of Mayflower descendants.
In New England, at first the Hinckleys would have been mostly Separatists.You can usually find records when they joined a church and sometimes when they left--especially if they left to join a different religion.