Why must an organisation represent the idea of an individual, rather
than the organisation acting in its own capacity while the individual
simply sits as its head? . . .
What function does this notion of a corporation sole serve?
A corporation sole is a way of organizing property ownership for an entity that predates the full development of the idea of an entity as a legal person that was developed primarily in Roman Catholic and Anglican canon law before business corporations came into existence, and is a label that was retroactively applied to this arrangement once the concept of a corporation became widespread.
In a corporation sole, ownership of property and other legal rights and obligations are vested ex officio in whomever holds a particular office. This was a compromise, similar to the compromise involved in the proto-corporate concept of a "trust" that formally vests ownership of property in a human being serving as a trustee that automatically transfers to a successor trustee. A corporation sole vested property ownership in a human being at a time when it wasn't obvious that there could be such as thing as an entity that was a legal person and sole proprietorships were the norm in the business world, while allowing for continuous succession of that property with the organization while keeping organizational assets separate from personal assets of the person holding the office in question.
Another purpose of a corporation sole which has contributed to its ongoing use in contexts like the Roman Catholic church is that each corporation sole has limited liability, so when each geographic region or function of the overall organization led by a different official has title to property vested in a corporation sole with limited liability, liability that, for example, makes only the property owned by the corporation sole of on Archbishop liable for that obligation rather than making the entire church denomination responsible for it. It is functionally equivalent to having many wholly owned limited liability subsidiaries of the larger organization.
Since a corporation sole vests the rights and obligations of this legal person in an office within some organization, however, rather than in the holders of transferrable stock ownership in the organization like an ordinary corporation, it doesn't really make sense for a commercial enterprise, as opposed to a non-profit or a government entity. But the non-transferability and non-alienability of ownership of a corporation sole is a feature rather than a bug within an organization that is contemplated to endure forever.
Indeed, a corporation sole is a close cousin of the civil law concept of an usufruct, which is a right in property that conveys the right to use the property and utilize its profits, but not to transfer or alienate the property. The difference is that an usufruct may be used to benefit the usufruct holder personally, while the use of the property of a corporation sole and the use of the profits and proceeds from the property of a corporation sole may only legitimately be used to further the ends of the office with which the corporation sole is associated.
As noted above, a corporation sole even an even closer cousin to vesting ownership of property in a trustee, solely for the purposes of the trust, which can be conveyed by operation of law when a successor trustee takes office. The only difference is that the limited liability aspect of a corporation sole for which the personal assets of the office holder are not responsible is explicit, absolute, and up front in a corporation sole, while it is only implicit and not quite absolute, in the case of the trustee of a trust.
The use of this relict form of organization for a newly created government office in the U.K. is probably an instance of imitation of the organizational details of some other government office in the U.K.
Some government offices in the U.K. were organized as corporations sole in imitation of Roman Catholic and Anglican corporations sole, in the era of U.K. history before the concept of ordinary corporations and governments as legal persons was well established and widely accepted. These ancient U.K. offices, like the Chancellorship, then continued to have this organization form out of inertia.
As result, the corporation sole form of organization continued to be an option on the menu list of how government division in the U.K. could be organized. Someone creating a new government division in the U.K. then decided to copy the corporation sole organizational model, rather than using another model for some reason.
The reason for choosing a corporation sole option rather than some other organizational option, may have been that the predecessor division of the government office that handled the functions of the newly created government office before it was established as a separate division, was organized in that manner. This would, for example, allow the new government office to almost seamlessly copy the organizational documents of its predecessor agency in a thoughtless way without doing any harm and thus saving time and money in the process of organizing the new agency. But I don't know this to be true for a fact.