What actually goes wrong with the book on the table
It asks a lot of a guest: stand up mid-course, walk to a table by the door, compose something meaningful on the first attempt with somebody waiting behind you, and write it legibly with a pen that has been in forty hands. Most people see the line and decide to come back later. Later is when the dancing starts.
So the book fills with the same four words from the guests who happened to pass it early, and the friend who flew in for the weekend — the one whose message you actually wanted — never signs it at all. Then somebody has to remember to take it home.
Doing it from the phone removes every one of those obstacles. A guest leans back in their chair between the main course and dessert, takes as long as they want over what to say, and sends it without moving. The ones who would rather not write at all can just talk to the camera for twenty seconds instead.
The messages end up in the same gallery as the photographs, so there is nothing to reconcile afterwards: the picture of your friend at the table and the greeting she recorded five minutes later are in the same place, under the same name.