You make some very worthwhile observations. From my perspective, the extensive similarities of the various texts is what provides the incentive to identify them. If the text were much more diverse, it would be difficult to find comparisons with a sufficient similarity index to pass the threshold, at which point the results would lose significance.

Any algorithm that gives less weight to the more common words will, in my opinion, be less able to handle this particular corpus and yield quality results. It is true that many of the keywords are used ubiquitously throughout the Bible: words like Lord, God, father, son, praise, faith, etc. There are also words that are much more common than in other types of text, such as inherit, blessing, perpetual, covenant, sacrifice, etc. And then, as has been pointed out, many of the words are of old-English form and would not be easy to accommodate with a stemmer: verb forms like hadst, hast, hath, gavest, giveth, rebuketh, etc.; pronouns like thee, thou, thine, and ye; and words like hence, hither, whither, whence, lest, whatsoever, whomsoever, etc. To add a little spice, there are the more unusual vocabulary words, like apothecary, winnowing, and all the wonderful names like Belteshazzar, Nebuchadnezzar, Abaddon, Apollyon, Zaphnathpaaneah, Mahershalalhashbaz, Sennacherib, etc. There are some interesting features of the language, along with some of the more repetitious aspects.

The entire corpus contains close to 13,000 words--not far from the average vocabulary of a high-school graduate these days.

A list of all the words in the KJV Bible, showing the number of occurrences for each, can be found HERE. It helps one to understand the peculiarities of this particular lexicon.