Great question. I don't know why I haven't ever thought to write about this; a stable (or permanent) state of life is a core element in understanding the distinction between consecrated eremitical life and lay eremitical life. I am very grateful you asked this. I checked it out online and as you said, while it was part of every accurate definition of consecrated life (including consecrated eremitical life) there isn't much written about it that I could find. So let me try to make explicit what has been implicit in my writings on this and related topics.
Stable in this context means lasting, solid, established, and (relatively) secure. The necessary noun "state" means ä fixed and permanent mode of life, established (in and by the Church) to acquire or practice a certain virtue (e.g., perfection in the Christian Life, holiness, the evangelical counsels within religious life, etc). Implicit in these definitions (and often explicit in canon law and the CCC) when the two words are combined is that such a stable state signifies an instance of a recognized way God is working in the Church, ecclesial approval and mediation of God's call, canonical standing (standing in law), appropriate oversite, support, freedom, and governance (legitimate superiors), and a formal (legitimate or canonical) commitment (say, to God via the evangelical counsels, for instance) by the one assuming the rights and obligations of the given state of life. The elements required for something to be considered a stable state of life tend toward structuring and extending to the individual life the elements necessary to truly pursue the given vocation in the name of the Church (and so, as a recognized representative of the vocation) with which the Church is entrusted. The Church recognizes several such states : Baptized or Lay, Married, Consecrated (Religious, Hermits, and Virgins), and Ordained. All require public commitments, whether Sacramental (Marriage and ordination) or via canonical profession and consecration (Religious, consecrated hermits, consecrated virgins).
When we begin to think about what makes a state of life in the Church a stable state we begin to understand why it is private vows per se never constitute the means to initiation into the consecrated state of life. They can be a significant part of the stable state of life we know as the baptized or lay state however, and they serve as significant (meaningful) specifications of one's baptismal consecration in this way. But in this case it is one's baptismal consecration into the lay state which defines one's stable state of life; private vows are expressions of that particular consecration but do not initiate one into it. Hence my references in many places to "lay hermits" --- hermits who live their vows in the baptized or lay state alone. In any case, private commitments, though often witnessed by a priest or spiritual director, are not actually received in the name of the Church or overseen by anyone in a formal or canonical way. There are no additional public rights or obligations, nor approved Rule the living out of which the Church as a whole is responsible for governing and supervising. Neither is there any process of mutual discernment by which one may be evaluated as to their capacity and suitability to assume the public rights and obligations of a given state (here I am thinking of the consecrated state), nor of methodical formation with such commitments.
Moreover, private vows are easily dispensed precisely because of their private nature. In other words one may make private vow as a hermit (whether with serious thought or on a relative whim) one day and days later (perhaps rightly, perhaps not) decide one has made a mistake or circumstances may change which make the vows inconvenient or an obstacle to a greater or more fundamental call from God re one's lay state. The vows can be dispensed by one's pastor. Because of the lack of oversight, etc.. other problems can creep in. If the person does not decide they have made a mistake an individual living a private dedication to eremitical life, for instance, may decide to substitute their own private notions of eremitical spirituality, or live inconsistently given conditions of health, education, training, economics, etc. Even for the most sincere and well-intentioned individual, in a private commitment there is no authority to whom the individual is canonically answerable, no canonical constraints or ecclesial vision to which one has committed oneself to make sure the hermit in this case can make, has made, is keeping, and continues to (be empowered to) keep through the years an appropriate and maturing commitment which the Church herself could recognize as consistent with the eremitical tradition and as rooted in the Gospel of Jesus Christ. Canonical standing provides a context which is stable.
Remember that consecrated persons act (live this vocation) in the name of the Church (and also their founders and spiritual Tradition) and that gives the People of God their own rights and reasonable expectations about the quality of life being lived by the person who has been professed and/or consecrated. The people also have a right to turn to the person's legitimate superior if there are grounds for suggesting the vocation is being lived badly or there are scandalous or concerning circumstances involved. Of course this is true only because canonical vocations are public vocations. But think how important it is that such expectations and accountability add to the stability of genuinely consecrated vocations! Accountability itself is a central element of a stable or permanent state of life. It shapes the vocation, challenges and supports it. In a public (canonical) vocation where the vocation "belongs" first of all to the Church who is entrusted with this calling, and only secondarily to individuals called by God through the mediation of the Church, stability is a function of clear channels of authority and accountability. This does not mean these channels are heavy-handed, of course, but it does require them nonetheless.
One of the things I appreciate most about canonical standing is the way
it establishes a person (or a community) in a living tradition in a way which means there is a clear and responsible dialogue ongoing between the individual, the Church, and the spiritual tradition involved. (This is true in religious families like the Franciscans, Dominicans, Trappist(ine)s, Benedictines, Camaldolese, etc. and it is true in eremitical life per se.) The continuing give and take as the consecrated person is granted and assumes a defined place in the living stream of eremitical tradition is tremendously edifying. The individual is formed in a given strand of the tradition and at the same time she will shape and extend the tradition with her own life. Edward Schillebeeckx writes about this powerfully in his essay on being a Dominican in God Among Us. A life that assumes this kind of responsibility, accountability, humility, and obedience has been initiated into a stable state of life that extends both behind and after her. She has taken a place within it and lives in a conscious and recognizable dialogue with and for this traditional thread, a thread which may have existed for two thousand years and stretches into whatever future the Church has. Private commitments which of their nature are truly entirely private (as opposed to public in the technical sense I use it throughout) simply do not do this.
The Church is a complex living reality. States of life within the Church have been some of the primary ways the Gospel has and continues to be proclaimed and ministry carried out; they are capable of being flexible and responsive to the needs of the world as a whole because they are also well-founded and rooted in a living tradition. Because of their stability (again, they are mutually discerned, publicly committed, ecclesially consecrated, governed and supervised) they can represent a way of life in away which teaches and inspires. When the congregation or individual requires assistance, when congregations reach the end of their natural life, for instance, canonical standing allows for various creative ways to be sure their life and/or charism can be handed on and, eventually, their history entrusted to archives so scholars can research them and allow their life, a response to the Holy Spirit in a variety of circumstances, to be of continuing benefit to the Church and world.
With regard to the lives of diocesan hermits or publicly professed vs privately vowed hermits I think you can see where the Church will be able to follow and assess the phenomenon of solitary eremitical life beginning in the late 20C. She will be able to look at the Rules written by c 603 hermits, interview bishops professing and supervising them, speak with their delegates, parishes, and dioceses, and just generally provide the story of professed solitary hermits since 1983 according to c 603. Both as individuals and as a group these hermits will contribute to the eremitical tradition, to assessments of what formation was helpful or inadequate, to considering what time frames were associated with successful discernment and formation of eremitical lives, to considerations re protecting the hermit's requirements for support, modes and effectiveness of supervision, the place and nature of limited ministry in the lives of these hermits, and possibly -- to some extent -- the hermits' affect on their local church communities.
We will also more easily contribute to theologies of eremitical life that allow chronic illness as a witness to the way God's power is perfected in weakness, for instance, because some number of us are chronically ill and sought out eremitical life in part because of this. Because we are professed and consecrated into a stable (and public!) state of life, the witness value of our lives will take on greater import for the Church and world. Sometimes folks decry the canonical paper trail that is attached to the profession of the diocesan hermit; others treat it as merely pro forma and relatively meaningless. But the paper trail is a witness to and even part of the stability of the hermit's life and a key to appreciating and researching eremitical tradition not only in the 20-21C but in comparison with it throughout history.