The C2 wiki presents only a very abbreviated version, and the chapter in question actually tells a whole bunch of stories. Here's a reasonably full excerpt:

Old man Cometto added that life is full of customs whose roots can no longer be traced: the color of sugar paper, the buttoning from different sides for men and women, the shape of a gondola’s prow, and the innumerable alimentary compatibilities and incompatibilities, of which in fact the one in question was a particular case: but in any event, why were pig’s feet obligatory with lentils, and cheese on macaroni.

I made a rapid mental review to be sure that none of those present had as yet heard it, then I started to tell the story of the onion in the boiled linseed oil. This, in fact, was a dining room for a company of varnish manufacturers, and it is well known that boiled linseed oil has for many centuries constituted the fundamental raw material of our art. It is an ancient art and therefore noble: its most remote testimony is in Genesis 6:14, where it is told how, in conformity with a precise specification of the Almighty, Noah coated (probably with a brush) the Ark’s interior and exterior with melted pitch. But it is also a subtly fraudulent art, like that which aims at concealing the substratum by conferring on it the color and appearance of what it is not: from this point of view it is related to cosmetics and adornment, which are equally ambiguous and almost equally ancient arts (Isaiah 3:16). Given therefore its pluri-millennial origins, it is not so strange that the trade of manufacturing varnishes retains in its crannies (despite the innumerable solicitations it modernly receives from kindred techniques) rudiments of customs and procedures abandoned for a long time now.

So, returning to boiled linseed oil, I told my companions at table that in a prescription book published about 1942 I had found the advice to introduce into the oil, toward the end of the boiling, two slices of onion, without any comment on the purpose of this curious additive. I had spoken about it in 1949 with Signor Giacomasso Olindo, my predecessor and teacher, who was then more than seventy and had been making varnishes for fifty years, and he, smiling benevolently behind his thick white mustache, had explained to me that in actual fact, when he was young and boiled the oil personally, thermometers had not yet come into use: one judged the temperature of the batch by observing the smoke, or spitting into it, or, more efficiently, immersing a slice of onion in the oil on the point of a skewer; when the onion began to fry, the boiling was finished. Evidently, with the passing of the years, what had been a crude measuring operation had lost its significance and was transformed into a mysterious and magical practice.

Old Cometto told of an analogous episode. Not without nostalgia he recalled his good old times, the times of copal gum: he told how once boiled linseed oil was combined with these legendary resins to make fabulously durable and gleaming varnishes. Their fame and name survive now only in the locution “copal shoes,” which alludes precisely to a varnish for leather at one time very widespread that has been out of fashion for at least the last half century. Today the locution itself is almost extinct. Copals were imported by the British from the most distant and savage countries, and bore their names, which in fact distinguished one kind from another: copal of Madagascar or Sierra Leone or Kauri (whose deposits, let it be said parenthetically, were exhausted along about 1967), and the very well known and noble Congo copal. They are fossil resins of vegetable origin, with a rather high melting point, and in the state in which they are found and sold in commerce are insoluble in oil: to render them soluble and compatible they were subjected to a violent, semi-destructive boiling, in the course of which their acidity diminished (they decarboxylated) and also the melting point was lowered. The operation was carried out in a semi industrial manner by direct fire in modest, mobile kettles of four or six hundred pounds; during the boiling they were weighed at intervals, and when the resin had lost 16 percent of its weight in smoke, water vapor, and carbon dioxide, the solubility in oil was judged to have been reached. Along about 1940, the archaic copals, expensive and difficult to supply during the war, were supplanted by phenolic and maleic resins, both suitably modified, which, besides costing less, were directly compatible with the oils. Very well: Cometto told us how, in a factory whose name shall not be uttered, until 1953 a phenolic resin, which look the place of the Congo copal in a formula, was treated exactly like copal itself— that is, by consuming 16 percent of it on the fire, amid pestilential phenolic exhalations—until it had leached that solubility in oil which the resin already possessed.

Here at this point I remembered that all languages are full of images and metaphors whose origin is being lost, together with the art from which they were drawn: horsemanship having declined to the level of an expensive sport, such expressions as “belly to the ground” and “taking the bit in one’s teeth” are unintelligible and sound odd; since mills with superimposed stones have disappeared, which were also called millstones, and in which for centuries wheat (and varnishes) were ground, such a phrase as “to eat like four millstones” sounds odd and even mysterious today. In the same way, since Nature too is conservative, we carry in our coccyx what remains of a vanished tail.

Bruni told us about an episode in which he himself had been involved, and as he told the story, I felt myself invaded by sweet and tenuous sensations which later I will try to explain. I must say first of all that Bruni worked from 1955 to 1965 in a large factory on the shores of a lake, the same one in which I had learned the rudiments of the varnish-making trade during the years 1946–47. So he told us that, when he was down there in charge of the Synthetic Varnishes Department, there fell into his hands a formula of a chromate-based anti-rust paint that contained an absurd component: nothing less than ammonium chloride, the old, alchemical sal ammoniac of the temple of Ammon, much more apt to corrode iron than preserve it from rust. He had asked his superiors and the veterans in the department about it: surprised and a bit shocked, they had replied that in that formulation, which corresponded to at least twenty or thirty tons of the product a month and had been in force for at least ten years, that salt “had always been in it,” and that he had his nerve, so young in years and new on the job, criticizing the factory’s experience, and looking for trouble by asking silly hows and whys. If ammonium chloride was in the formula, it was evident that it had some sort of use. What use it had nobody any longer knew, but one should be very careful about taking it out because “one never knows.” Bruni is a rationalist, and he took all this very badly, but he is a prudent man, and so he accepted the advice, according to which in that formulation and in that lakeshore factory, unless there have been further developments, ammonium chloride is still being put in; and yet today it is completely useless, as I can state from firsthand experience because it was I who introduced it into the formula.

The rest of the chapter is Levi describing how he had been assigned the task of investigating why a whole pyramid of paint mix was screwed up, when the paperwork looked all correct. It turns out that instructions for '2 or 3 drops' had gotten dirty and started to look like '23 drops', both ruining the paint and fooling the chemical tests; he discovers a clever way to fix the stockpile, which uses the ammonium chloride to neutralize the responsible chemical and which is added as a preventive; except a decade later, they stopped using the responsible chemical at all; thus, then the ammonium chloride kept being added entirely unnecessarily:

Since the storeroom contained several shipments of perilously basic chromate, which must also be utilized because they had been accepted by the inspection and could not be returned to the supplier, the chloride was officially introduced as an anti-livering preventive in the formula of that varnish. Then I quit my job: ten years went by, the postwar years were over, the deleterious, too basic chromates disappeared from the market, and my report went the way of all flesh: but formulas are as holy as prayers, decree-laws, and dead languages, and not an iota in them can be changed. And so my ammonium chloride, the twin of a happy love and a liberating book, by now completely useless and probably a bit harmful, is religiously ground into the chromate anti-rust paint on the shore of that lake, and nobody knows why anymore.

On the whole, I read Levi's chapter as indeed anti-Chesterton's Fence. The legacy mechanisms are often useless, potentially highly wasteful (16% of the resin is wasted in that phenol story, for absolutely no reason), confusing, risk problems down the line (one can easily imagine the onion or ammonium screwing with later changes or ingredients, as Levi notes that it is apt to increase rusting), and can obstruct improvements (if you use the onion unthinkingly, you will be less inclined to get a better way of measuring temperature like a thermometer). Investigation into anomalies or blackboxes may be expensive, and can require deep theory of little apparent practicality, but can pay off big (like rescuing an entire stockpile of paint from futility) and should be done and as much rendered legible as possible. And if one doesn't wind up able to present a complete history and explanation as a nice little case-study that Levi can write up, because it's that dumb a thing (the smudged recipe having already been thrown out, say), then oh well, one should throw out the extraneous thing and investigate it that way, with no particular respect for the onion, which deserved no special veneration or status and was exactly as foolish as it looked, our fathers being no wiser, and often less wise (particularly in chemistry or technology), than ourselves.

Yeah, I see this the same way. In The Onion in the Varnish, the author notices that it's confusing that there's an onion in the varnish, but they don't immediately stop putting it in. They first investigate to try to find out why it's there. Chesterton's Fence doesn't say you can never remove fences, just that you should know why the fence is there before you do, and the person in the Onion in the Varnish does exactly that.