Much as I hate defending the Watchtower, I think you are stretching a bit here. In the gazillions of songs that have been written, it's almost inevitable that there will be similarities between short sections of some of them. In this case, one phrase of the song contains a similar progression. It's hardly the case that "Walking in Integrity, is a complete rip-off of the jazz standard, Stella by Starlight." It's not really true, as you state, that the two have the "same tune," except in one small part of each song. Even if the former was influenced by the latter, I don't think the similarity is of sufficient substance to complain about plagiarism or a "lack of integrity in its authorship." It's entirely possible that the composer of Walking in Integrity simply utilized the phrase subconsciously without realizing it.
Now, I'm not saying that the WTS wouldn't plagairize a song if they thought they could and thought it would be helpful to them, but the similarity in this case just isn't of sufficient scope to back up the accusation, in my opinion.