What I find appealing about Byzantine Priority is 1) it seeks the preserved historical text from an actual view of textual transmission, 2) is thoroughly consistent, not changing the entire paradigm for "how we discern the correct readings" on a case by case basis (the TR position is a form of "text-theory eclecticism" using radically different and contradicting argumentation depending upon the reading in question) and 3) is demonstrably the historical text, transmitted throughout the age of the church and not simply the late 16th century edition with readings (though admittedly very few) that are not in the historic stream.
Byzantine priority isn't based on an actual view of textual transmission The view itself is probationary. It provides a theory of textual transmission based on the similarity of readings in a family of mss. which have for that reason been designated "Byzantine." It presupposes the genealogical theory, which was never adequately established in the first place.
There is no promise that a family of mss. would be preserved. The promise is that the Word would be preserved for the church, and the church is much larger than the eastern tradition.
As for the TR, in a number of cases it ends up having a broader base of support because it is not restricted to the genealogical theory or a conjectural history to support it. It gives the text to which the church has historically appealed, and has a range of support which is not limited to mss.
What percentage of "manuscripts" has been preserved in real terms, not just in terms of relative value with other archaeological discoveries? The percentage is quite small in terms of the number of mss. which must have actually been produced. To posit a theory of "providential preservation of mss.," it seems requisite to have a good percentage of mss. in possession, which is just not the case. There are too many gaps to be able to say for certain that specific "texts" developed at definite times and places and bear a definite relation to other "texts." Most of it is still hypothesis. One should not call on "providence" to fill in the missing links.