Yes, the so-called "essential fatty acids" are, indeed, essential" (i.e. required, but unable to be manufactured by the human body.)
The question includes a link to the Wikipedia page for essential fatty acids which defines them (in humans) as : alpha-linolenic acid and linoleic acid.
The same page explains the history of the research into them, describing a number of experiments that identified them as essential.
In the 1950s Arild Hansen showed that infants fed skimmed milk developed essential fatty acid deficiency. It was characterized by an increased food intake, poor growth, and a scaly dermatitis, and was cured by the administration of corn oil.
Later work by Hansen randomized 426 children, mainly black, to four treatments: modified cow's milk formula, skimmed milk formula, skimmed milk formula with coconut oil, or cow's milk formula with corn oil. The infants who received the skimmed milk formula or the formula with coconut oil developed essential fatty acid deficiency signs and symptoms.
[...] patients undergoing intravenous nutrition with glucose became isolated from their fat supplies and rapidly developed biochemical signs of essential fatty acid deficiency (an increase in 20:3n-9/20:4n-6 ratio in plasma) and skin symptoms.
The Wikipedia page pretty much answers the question, but I followed up a couple of sources, so I wasn't only relying on unreliable secondary sources.
This paper described three case studies of patients malnourished by a deficiency in linoleic acid. It explained they suffered "skin lesions, characterized by dryness and scaly appearance" until they were administered with Intralipid that contains linoleic acid.
It appears that the daily requirement for linoleic acid in the adult, particularly during the period of rapid anabolism, has not been clearly established.
This analysis of recommended dietary intake from 2004 summarises some of the literature.
Prospective cohort studies examined the effect of the intake of dietary fat and ω3 PUFA
from plants on coronary heart disease in humans. [...] After adjustment for
non-dietary risk factors and total fat intake, intake of ALA was significantly negatively
correlated with risk of myocardial infarction (relative risk 0.41 for a 1% increase in
energy from ALA, P < 0.01).