An impending wedding is usually one of the most publicized events in any city. The banns must be published, typically in a special section of the newspaper. In order to give notice for anyone who may object or know about a prior bond. Also any hint of duress or urgency that may impede free consent. The banns are the actual execution of the ceremonial “callout” you see in films.
The witnesses of a wedding are not optional. The witnesses serve as representatives of the general public. Typically a clandestine wedding would be invalid without witnesses to verify and vouch for the identity, presence, and consent of bride and groom.
Taking photos for verification is sort of after-the-fact, and it would be most unfortunate for the banns to miss the mark until after the ceremony, or the consummation.
But only crazy people would consider a wedding ceremony “private” or “closed to the public” other than “renting an officiant” and flying off to a Caribbean elopement that only your billionaire girlboss bridesmaids can afford.