The Republican party was religious radicals who either requested or demanded a new way and fought the status quo slavery loving (social conservative) democrats.
People in current seats have nothing to do with the history of the organizations they are in and if transported to the past would often more welcome in opponent organizations. This is also true of the revolutionary founding fathers and their very different views and make up than the puritans.
All indications are that the puritans could neither write not respect a constitution. Rhode Island was made up of people who wanted freedoms not stifled by that bunch.