I think that the aversion to the word "basically" you describe in your professors comes down to its differences to "essentially" in semantic range and contextual usage.
"Basically," according to Merriam-Webster, means "at a basic level; in fundamental disposition or nature." To me it carries with it the meaning of roughness, of generalisation, and so has a meaning close to "I am ignoring some things, but this is good enough," or put simply, "here is the overall picture." I would assume that it has thus developed negative connotations (for the user of the word, not the referent) in an intellectual or academic context because it is a word that lacks specificity, precision, and so exposes to some degree poor communication skills.
Perhaps the etymology of the word "basically" doesn't help, cognate with "base," which of course was used to describe something undignified or animalistic. Today, the word is also used in casual social settings, and this might contribute to the chagrin of your professors, who of course would want formal writing to be different in register to casual parlance.
"Essentially," in the Merriam-Webster entry for "essential" (sense 1), means "of, relating to, or constituting essence; inherent." I rephrase that as "in the most necessary form that I have identified," where "necessary form" catches the conditions that must be in order to make something such. If poetry essentially is figurative language (as Professor Harold Bloom asserts in The Art of Reading Poetry), then poetry cannot be poetry without figurative language, at least in Bloom's perspective. If I say that Bach is essentially harmonic, then it follows that Bach is not Bach without harmony. These are only two instances, but I think in most cases, the word "essentially" therefore actually has precision and specificity.
And consider how the variants of "essentially," such as the word "essence," have been used in philosophy. I can think of its uses in Plato, Hegel, Kant, where its ontological meaning is employed. It might be that your professors inherent this intellectual tradition, and so like the philosophical properties that have accumulated around the word (or are these properties part of the word's essence?)
Although "essentially" is also used in casual conversation, my own experience leads me to think that it is less commonplace, and perhaps employed by more highbrow members of society.
Ultimately, I guess that you could say that there is a difference between simplistic and simple, which reflects the difference between basically and essentially.
So I think there is reason for privileging one word over the other because of the semantic differences, however nuanced, between the two. In formal writing, you want the kind of precision that "essentially" can give you. If there is any pretentiousness on your professors's part, I'm guessing this would be to ensure your formal writing isn't misread as something casual. I don't think there's too much academic snobbery at play.
(Here is the original opening paragraph I wrote, but I realised it was quite convoluted and I ended up losing myself in it, but it might have some points for consideration:
Certainly both words have a similar meaning, and as you say both try to reduce or explain complexity in the most accessible or fundamental terms. But "basically" perhaps is too reductive and simplistic, stripping something down to the level where it is less evolved, and so fails to communicate properly what you are referring to. "Essentially" does not divest, but invests, showing something's most necessary properties to exist. "Basically" is reductionism, "essentially" is (perhaps tautologically) essentialism.)
PS I really struggled to not use both words as I was writing this! Hope it helps!