The idea of the label being repeated is that the reader will know, no matter what the format of the table, that it is continued. You don't actually need the running label on the bottom of the first page of the table, just succeeding pages.
As to your question about why only sequentially numbered tables have this requirement, I really can't think of a good reason. It would make sense to me that all tables have the running label if they are continued.
Sorry for the delayed response. I was out of town. The attached pdf is a chapter of the book. Table 14.1 on page 360 is a table that is a full page of text. The attached Braille 2000 file is the braille for the table.