So much angst over what appears to me to be a more than adequate approach.
Just a couple of comments:
The conversion to a metric calculation may be of use to those who have no access to traditional English units, but is an example of where the purity of metric actually complicates the calculation. While the metric version eliminates confusion between pounds mass and pounds force, in this simple non-accelerated system there is no difference, hence no confusion. As pointed out in several prior posts the answer is a the simple product of area times tensile strength in traditional units.
I don't really think a proof test is necessary, but you have seen the variety of opinions here. Getting a solid theoretical answer will be more difficult than getting empirical results.
You haven't mentioned any interface to the wall. I would recommend some form of attachment to eliminate any possibility of a rattle or bang from either air currents or vibration from passing traffic. Such an attachment greatly complicates the theoretical analysis, but the need has often been demonstrated in practice. There are ways to make said attachment theoretically sound (applies a load to the book case without supporting any vertical loads), but only one is worth considering in practice, which is placing the ceiling end of the threaded rod slightly closer to the wall than the penetrations of the shelves. The problem with this approach is that I am not aware of any really good way to determine how much closer you need to be. You can do the geometry to see how much side load is generated, but the data for how much is enough will be non-existent.
Or you can do what most tradesmen would do and put a couple of wall anchors in and screw it to the wall. The reality is that these anchors will deform as needed under the vertical loads and it will work.