I don’t know the answer to this question. Let me suggest some possibilities:
Perhaps the Brethren are worried that publishing the Handbook will encourage people to treat it as a legal text. There are two possible problems with this. It might then encourage people to use deviation from the Handbook to attack priesthood leaders, when the Handbook is merely intended to orient them in particular ways not necessarily limit their ability to deviate. Alternatively, treating the Handbook as a legal text might discourage members from approaching issues prayerfully and flexibly rather than legalistically. By keeping it private, the thinking might go, we limit its public authority as a text and thus limit legalism. The problem, it seems to me, is that it tends to function as a legal text anyway, just a problematic one because it is not public.
Perhaps the Brethren are worried that the deep secrets of the Church will be revealed if the Handbook is made public and deeply embarrassing things will come to light. The problem with this is that there just isn’t anything particularly scandalous in the Handbook. I’ve read it. There’s no deep secrets in it. Furthermore, copies of it are available in various research libraries — including BYU Special Collections — to say nothing of the Internet. If it contains deep secrets they are already out of the bag. Remember all of those breathless investigative reports in the NYT exposing the way that Ward Clerks are supposed to prepare quarterly reports or the operation of the Bishop’s Youth Council? Yeah, me neither.
The only reason that really makes sense to me is legal, and even that doesn’t really make sense to me. There are a number of cases involving litigation against employers in which corporate manuals held out to the public or to employees were deemed to be contracts or the very least promises on which one might reasonably rely. Accordingly, they were made the basis of lawsuits. Because contractual liability is strict rather than fault based, showing that a manual is a contract is potentially really powerful because you can impose liability simply by showing failure to comply with procedures. Importantly, one needn’t show negligence or malice in failing to comply with procedures, so meeting the evidentiary requirements for liability are easier. Although, on the back end one’s remedies in contract cases as opposed to situations involving fault-based liability are — to use technical legal language — less umphy. No punitive damages for breach of contract, for example. Perhaps the Church wants the Handbook kept private in order to head off suits by members and others based on some local leader’s failure to comply with procedures in the Handbook.
The problem with this last explanation is that lots of jurisdictions don’t follow the handbooks-are-contracts theory. To again use technical language this is what is known among lawyers and legal scholars as some-batshit-crazy-thing-that-state-courts-in-California-and-New-Jersey-do-sometimes (SBCTTSCICNJDS, pronounced spects-cinc-noj-dos; there is a dispute as to whether the stress is on the second or third syllable). More importantly, it’s, as I understand it, a pretty recent development, and the CHI has been kept confidential for some time. (Although its worth pointing out that what became one of the main 20th-century versions of the CHI was published by the Church in the 1930s as a lesson manual and later as a book, namely John A. Widsoe’s Priesthood and Church Government. Go figure.)
So there you have the anatomy of my ignorance. I don’t know why the CHI (or at least the first volume of the CHI) is not public. The only thing that really makes sense to me is very cautious lawyering, but I suspect that historically this isn’t the actual explanation.