The strongest endorsement that I could find from a UK English source in favor of using 's after singular nouns of any kind to indicate possession is this brief treatment from The Oxford Guide to Style (Oxford University Press, 2002):
Use 's after singular nouns and indefinite pronouns that do not end in s:
[Examples:] the boy's job, the BBC's policy, nobody's fault, the court's decision, the bee's knees, one's car, Oxford's bells, Mary's garden, a week's time, Yasgur's farm
As this guideline suggests, the situations in which re-forming an expression from "X's Y" to "the Y of X" has the most deleterious consequences are the ones involving idioms: "the knees of the bee" and "the meow of the cat" sound ridiculous—and indeed foreign—when rendered thus. But so does "the time of a week" (or to a lesser extent, "the span of a week" or "the duration of a week") in place of "a week's time," or "the thought of a moment" for "a moment's thought."
Another treatment, with fewer nonhuman examples of the 's treatment in action is from the "English Grammar Today" section of Cambridge Dictionaries Online:
We use apostrophe s (’s), also called possessive ’s, as a determiner to show that something belongs to someone or something:
Is that Olivia’s bag?
Britain’s coastline is very beautiful.
We can also use it in complex noun phrases:
Greg is her youngest daughter’s husband.
We can use two possessive ’s constructions in the same noun phrase:
We went to Jake’s father’s funeral.
We also use possessive ’s to talk about time and duration:
Is that yesterday’s paper?
I’ve only had one week’s holiday so far this year.
A bit later in the same discussion, the Cambridge guide points out a feature of English in an accurate but potentially misleading way:
We don’t usually use the possessive ’s with things:
the door handle
the door’s handle
the shop window
the shop’s window
the kitchen table
the kitchen’s table
But native English speakers who blithely say "the bee's knees" and "the cat's meow" wouldn't say "the door's handle" or "the kitchen's table" in any event, because the nonpossessive form is sufficient; for the same reason, they wouldn't be inclined to say "the handle of the door" in preference to "the door handle," and they certainly wouldn't say "the table of the kitchen" in preference to "the kitchen table."
However, the Cambridge Dictionary page doesn't stop there. Farther down the page it offers these guidelines:
’s or of or either?
There are some general rules about when to use ’s and when to use of but there are many cases where both are possible:
The film’s hero or The hero of the film
The car’s safety record or The safety record of the car
The report’s conclusion or The conclusion of the report
Sometimes when we first mention a noun, we use of, and later when we refer to it again, we use ’s:
The mountains of Pakistan are mostly in the north. At least one hundred of them are above 7,000 metres … Most of Pakistan’s mountains are in the spectacular Karakoram range.
When we don’t use ’s
We don’t use ’s when the noun is not a person, animal, country, organisation, etc., or when the noun phrase is very long:
The name of the ship was ‘Wonder Queen’. (preferred to The ship’s name was ‘Wonder Queen’.)
The house of the oldest woman in the village. (preferred to The oldest woman in the village’s house.)
When we don’t use of
When we are talking about things that belong to us, relationships and characteristics of people, animals, countries, categories, groups or organisations made up of people, we usually use ’s:
The men’s dressing room is on the left at the end of the corridor.
The dressing room of the men …
The cat’s paw was badly cut.
The paw of the cat …
This block of advice is exceedingly difficult to make sense of. In one section we read that "film's hero" and "car's record" and "report's conclusion" are all okay, despite their not being a person, animal, country, or organisation, and in the very next section we're informed that "We don’t use ’s when the noun is not a person, animal, country, organisation, etc., ..." Either the etc. at the end of that phrase is large enough to comprehend works of art, machinery, and publications (among other possibilities) or the guidelines aren't paying attention to one another.
Clearly, even Cambridge's online grammar guide doesn't endorse your professor's rule limiting the application of 's possessive forms to people. But it does muddy the water sufficiently that a determined professor might find some solace (if not vindication) there.
To me, the greatest harm that an endless series of "the Y of X" constructions does to a lengthy piece of writing, when used in place of a mixture of "the Y of X" and "X's Y" elements, is to make the writing sound needlessly wooden and rote. It's easy to underestimate the contribution that varied elements make to an essay—until you read one that uses the same restricted palette of constructions over and over.
The Oxford Guide to Style puts no restriction on what sorts of singular nouns can take 's to yield a possessive form. The Cambridge guidelines, though all over the place, at least indicate that significant areas of nonhuman things can be assigned 's possessives. I would be inclined to show both the Oxford treatment and the Cambridge treatment to the professor and ask him what he made of them. Even a shift from his current person-only rule to Cambridge's person, animal, country, organisation, etc.-only rule would give you some latitude in concluding that a plant, a node, and a network stack, for example, were all covered by the "etc." term of the latter rule.