The only sane reason I can think of for giving in to your professor's preference concerns bibliographies and, specifically, bibliographies created with BibTeX and a bibliography style that sorts entries alphabetically by authors' surnames. (Aside: The issues raised in the remainder of this answer do not pertain to bibliographies generated with biblatex+biber.)
For deeply historical reasons, BibTeX does not sort "accented" characters such as á, ä, à, and â that may occur in the author and editor fields together with the letter A; instead, they are treated (for sorting purposes only) as coming after the letter Z. This affects how pieces authored by, say, Rädermacher, Ràdon, Rámos, and Râmuz [with some deliberate misspellings -- sorry!!] are sorted relative to pieces written by, say, Randall and Rybczynski. Do you want these entries to be placed before Randall or after Rybczynski? You may expect the former to happen, but BibTeX delivers the latter outcome.
Here's another example: Suppose your bibliography contains single-authored entries by Hasbrouck, Haščič, Hase, and Hayworth. Would your expect Haščič's publication to be listed before Hase's -- or after Hayworth's? If you enter the author's name as Haščič instead of as Ha{\v s}\{v c}i{\v c}, BibTeX delivers the second option.
For more information on how to enter accented characters in BibTeX entries in order while avoiding the sorting-related issues noted above, please see the posting How to write “ä” and other umlauts and accented letters in bibliography? [Shameless self-citation alert!]
Well, I can think of a second reason: If your computer keyboard does not provide a straightforward method for entering certain accented characters -- not just á, ä, and â but also, say, angstroms, ogoneks, and thorns [!] -- it's obviously very nice to know that you can enter them as \'a, \"a, \^a, etc as well.
A separate comment: If you compile your document with pdfLaTeX and if you input accented characters directly into your document, I will assume that you also load the fontenc package with the T1 option. If you compile your document with XeLaTeX or LuaLaTeX, there's no need to load the fontenc package.
Addendum: The following MWE, compiled with pdfLaTeX on a MacTeX2020 system, demonstrates that BibTeX places entries by Rädermacher, Ràdon, Rámos, and Râmuz (note the accented characters) after Rybczinski rather than before Randall. Ouch!! Hence, in order to obtain what most people would think is the "correct" sorting outcome, it is necessary to enter these names as R{\"a}dermacher, R{\`a}don, R{\'a}mos, and R{\^a}muz if and when they occur in the author or editor fields of BibTeX entries.

\documentclass{article}
\begin{filecontents}[overwrite]{mybib.bib}
@misc{r1,author="Randall",year=3000,title="Thoughts"}
@misc{r2,author="Rädermacher",year=3000,title="Thoughts"}
@misc{r3,author="Ràdon",year=3000,title="Thoughts"}
@misc{r4,author="Rámos",year=3000,title="Thoughts"}
@misc{r5,author="Râmuz",year=3000,title="Thoughts"}
@misc{r6,author="Rybczynski",year=3000,title="Thoughts"}
\end{filecontents}
\usepackage[T1]{fontenc} % useful under pdfLaTeX
\usepackage[authoryear]{natbib}
\bibliographystyle{plainnat} % use a bib style that sorts entries alphabetically
\setlength\bibsep{0pt} % optional
\begin{document}
\nocite{*}
\bibliography{mybib}
\end{document}
[utf8 ){inputenc}if your latex version is no older than april 2018, since it is now the default encoding. But you still have to load theT1fontenc. – Bernard May 19 '20 at 18:53