The following is my attempt at the question after a very brief historical research. The history of the court system and of the regulation of legal professions in English is just... chaotic until late 19th century, so there will be many points that are incorrect or can be more precise below.
At the advent of legal system, advocacy was a different from representation. Barristers advocate at the bar under instruction; attorneys and solicitors represent their clients. At their roots, attorney means one appointed as an agent (e.g. power of attorney = power to act on behalf of someone), and solicitors are called solicitors because they solicit causes on behalf of a client.
Champerty and maintenance, or the encouragement of disputes by non-affected parties, were and remain (though with various modifications) in several jurisdictions illegal or contrary to public policy. The profession of lawyers, who are not parties to a dispute, is also concerned, even today (e.g. the controversy about contingency fees). Barristers were the original men who were learnt at (English) law and the King allowed them to give counsel as their profession against payment without being found guilty of maintenance or champerty. It was only in 1967 that a person other than a barrister may represent another person in a litigation before a court (although various exceptions existed) in England. At the same time, barristers considered themselves learnt men who should not practice laws only for the fees, but for the advancement of public good and justice; this is also why the lawyer fees are still sometimes called honorarium.
Attorneys and solicitors were not originally persons who learnt law "properly", at least in the eyes of the Inns, and could not represent their clients before a court without violating the prohibition against maintenance, as law was not their profession.
The high fees and limited number of barristers, as well as the explosion of commercial and industrial activities, made it attractive for people to consult "lesser" legal practitioners, such as those who were in the business of soliciting (acting as an agent in) causes for a master (e.g. instructing a barrister on behalf of a client). The Inns of barristers were not very happy about it; it should be noted that Inns are professional guilds that have an interest in having a monopoly over their profession (much like the case of many professions, including legal ones, in modern times).
But non-barrister legal practitioners became commonplace anyway and the regulation of these non-barristers became necessary. While initially solicitors were not seen as independent professional advocates, a limited right of audience was gradually granted.
As the regulation becomes necessary to protect the public, attorneys in common law courts, and solicitors in equity courts, were given the privilege of practicing the law in limited form by the courts themselves. An attorney or a solicitor was sworn and accredited as an officer of their respective court by a judge, whose court controlled the legal practitioners (other than barristers) in their court and regulated them in absence of royal or statutory intervention.
The courts maintained rolls of admitted attorneys and solicitors, which still exist today. The Roll of solicitors (the profession most attorneys were absorbed into) is now maintained by the Solicitors Regulation Authority (originally by the Master of the Rolls in the Chancery court, then by the Law Society, then by the SRA).
In England and Wales, solicitors today are subject to statutory regulations under the various Solicitors Acts (notably the 1974 one). They are statutorily officers of the Senior Courts who retain disciplinary power over the solicitors admitted to practice before them.
Barristers on the other hand are mostly self-regulated by the Inns. Although the regulation of barristers now has a statutory basis and subject to public supervision (from the Courts and Legal Services Act 1990 to Legal Services Act 2007), they are not as extensively subject to rules set by the Parliament or government as solicitors.
To summarize, the barristers always had the "vibe" of being the learnt men (and now persons of all genders) of law who are independent, honourable and capable of self-regulation, while the profession of solicitors developed from mostly unregulated legal service agents, who were eventually regulated at the motion of the courts in the way of requiring them to be sworn as officers of the courts (such to give the courts disciplinary powers over them).
With the development of modern legal system, the distinction between barristers and legal professions has become less clear in some jurisdictions and ceased to be relevant at all in others.
As to the reason why the distinction is still so much more pronounced in the UK than elsewhere. I would guess the following. The legal, justice and judicial systems in the colonies benefited from the fact everything was started from the beginning, without historical baggage of the several different English courts which were not consolidated until 1870s. And it is also easier to never afford such privileges in the first place, or to take them away without the historical baggage, than to take away existing privileges with historical claims in England. The general anti aristocratic attitudes in colonies probably also restricted the rather elitist tendency of the likes of the Inns of barristers. The governments were more willing to regulate the legal profession in its state capacity.
The articles Counsellors and Barristers. An Historical Study, Solicitors and the Law of Maintenance 1590-1640, and the book Introduction to English Legal History (chapter 10), were particularly helpful during my research.