According to this site, one of the questions asked at ordination is "do you have eczema?" to which the reply must be "No, Venerable" in order to ordain. Is this accurate? Is it the case the one with eczema cannot ever become a monk?
An explanation (including the origin story) of this rule is given on pages 194 through 195 of The Buddhist Monastic Code II -- The Khandhaka Rules Translated & Explained by Thanissaro Bhikkhu.
Eczema covers a wide variety of skin diseases, differing from those included under “leprosy” in that they are not debilitating and do not ulcerate or ooze. Thus ringworm and athlete’s foot would come under this category. As under the preceding category, small, non-spreading infestations in an area covered when fully robed would be allowable.
The degree of strict interpretation of such rules largely depends on the monastery, tradition and abbot under whom one ordains. Most monastic orders I know adapt these rules to suit their circumstance in practice.
In the link you've quoted a reasonable explanation is offered
There are bhikkhus who could enter the sangha exclusively to benefit from care by doctors who provide free health care to the bhikkhus. Others could enter to elude legal obligations. To avoid problems of this type, in the first part of the procedure the postulant is asked fifteen questions, which he must be able to answer satisfactorily in order to be accepted.
Pragmatism vs dogmatism is a common theme in all religions. The Buddha warns about falling into the textbook learning trap on several occasions. His teaching fundamentally is about right action stemming from right intention cultivated through direct insight.
In the order of Plum Village, established by Ven. Thich Nhat Hanh, all monks are covered by basic health insurance (as mandated by French law (h/t @chrisw)).
I'm guessing that such rules were the compulsions of "maintaining the health of the flock/herd by sacrificing the individual". In those ancient days when antibiotics and anti-fungal creams were undiscovered, the teachers had the duty of constantly ensuring the physical well-being of the monks and disciples in the monastery by eliminating contagious diseases. Such practical/physical problems and challenges may have taken precedence over philosophical concerns on a day-to-day basis, and such rules, though situational, may have been worded as absolutes.
We don't know the historical context when such rules were written, and one can only speculate. I am speculating that in some decades or centuries when skin diseases assumed epidemic proportions, these rules may have been written, and may have seemed like absolute imperatives at the time. Later on, the threat of skin diseases may have diminished somewhat, but, in the possible absence of a system within the Sangha to review, amend or cull the rules that had lost their need or context, such rules may have become frozen in time, and may even have acquired the halo of ancient-therefore-immutable.
I don't have any authorities to cite.