One of the main problems here is that in the original Order a letter of invitation was required to join the Second Order. Admittedly most members who reached Portal progressed on to 5=6, but not everyone was admitted. Crowley is the best example of this. He applied and was refused, and it was Mathers' overruling of the other Adepts decision that broke the camel's back and began the rebellion.
So, how could this element of "invitation only" be carried forward to self-initiation? The truth is: it can't. If I were to apply for 5=6 to myself, I am hardly going to refuse myself. I will think I'm ready even if I'm not. While it is hoped that the student has developed sufficiently to recognise their own progress and such by this stage, this is often not the case, especially with self-initiation, where there is often no mentor to point the way or correct mistakes in knowledge and progress, not to mention the possible emotional and psychological matters that may have arisen, such as the all-too-frequent rearing of the ugly head of Ego.
The requirement for 5=6 is that the candidate must be "Vaulted", which is to say that they must be initiated into a fully equipped (and hopefully accurate) Vault of the Adepti. It was the building of such a Vault that started much of the modern revival of the Golden Dawn. When we look at the 5=6 ceremony it is not difficult to understand why this Vault is needed and why a simple plain room will certainly not do. It is the very heart of the ceremony. The candidate discovers the mythical Vault of Christian Rosenkreutz, after the figurative 120 years, as described in the Rosicrucian manifestos. If we remove the Vault we remove a huge chunk of the ceremony - indeed, the entire setting of the initiation as a whole. Combine with this the loss of the specific energies charged on Corpus Christi, etc., and it makes for a petty barren substitute for the 5=6. So, if a student truly wanted to self-initiate into 5=6, the construction of the Vault would be needed, just as he or she is required to construct all the wands, lamens, and other tools and regalia for the Outer Order ceremonies (when following the Cicero instructions, that is).
Apparently Mathers wrote a self-initiation ceremony for 5=6 in the Alpha et Omega. I have not yet seen this, but I was told it was weak in comparison to the actual 5=6 itself. I cannot really comment on what I have not seen, but I think it does support the idea of self-initiation, although I am hesitant to consider what it would be like in comparison to a fully Vaulted initiation.
It should be noted that while self-initiation into Adeptus Minor may be impossible, that does not mean one can not become an adept in the broader sense. Aaron Leitch pointed this out by stating: "Let us not confuse the word "adept" with the "Adeptus" Grades in the HOGD", which is an important point to make. Regardie, for example advocated using the Watchtower ceremony to become an adept, but I do not recall him saying it made the person an Adeptus Minor (although that is arguably what he meant, given subsequent teaching, for example, to Cris Monastre). There are adepts in multiple traditions, but an Adeptus Minor in the Golden Dawn is a specific grade, and one does not equate with the other (true adepthood is not necessarily granted with the grade, for example). One can still study the gradework of Adeptus Minor, and become an adept in the larger sense of the word, but the consensus remains, even among those who advocate self-initiation into the Outer Order grades, that self-initiation into Adeptus Minor is simply not possible.
To end this topic, I will link again to my post on my Gnostic blog Henosis Decanus nearly two years ago, which deals with a parallel subject that frequently pops up in ecclesiastical Gnostic circles: self-ordination.