A clue to the reasoning can be found by looking to the example of Canada, which originally had a guarantor system quite similar to the UK's, but changed it greatly in 2007. From the "New Guarantor Policy" section:
Historically, two criteria have been used to assess the appropriateness of the list of professional or occupational groups eligible to act as guarantors. The first criterion is to ensure that all applicants have reasonable access to a guarantor, and the second is to ensure that the guarantor’s eligibility can be readily verified by Passport Canada. For verification purposes, Passport Canada attempted to obtain membership directories of those professional or occupational groups eligible to serve as guarantors in order to validate their eligibility as guarantors.
So the original reasoning was that it's easier to verify a person is who they say they are when they belong to some licensed profession that keeps careful track of its members. If you make up a fictitious dentist as your guarantor, it's more likely to be discovered than if you make up a fictitious retail clerk.
However, this policy ran into many of the problems you note in your question:
The requirement that guarantors be professionals continually generated complaints from the public who perceived the policy to be elitist, and from professionals who were not included in the eligible list. The evolution of professions has created many ambiguous areas, making eligibility difficult to determine, both for the public and Passport Canada. Many groups did not provide Passport Canada with membership directories, requiring Passport Canada staff to contact the association or licensing body to verify the guarantor’s eligibility.
For applications from within Canada, Passport Canada now only requires one's guarantor to be a passport holder; there is no requirement that they hold any particular profession. Additionally, family members were not eligible as guarantors under the old policy, but became eligible under the new policy. Passport Canada now uses its own records to verify the identity of the guarantor.
As it happens, a small vestige of the old guarantor-based system still remains. Passport Canada still allows applicants who are applying from outside Canada to use an occupation-based guarantor for their application. Presumably this is because Canadian passport holders are rarer outside of Canada, and a Canadian citizen who has lived in a foreign country for many years might not know any Canadian passport holders.