I guess I'm the canonical person to answer this question. I wrote those notes as a PhD student, a long time ago, to go along with a talk I was giving at a grad student conference. They were basically my lecture notes, written for myself. I have learned a lot since then, and would probably write things differently today. While having a model structure does allow you to solve the size issues, it is not necessary for doing so. In answering this question, I learned that a similar question, also pointing to my old seven-page lecture note, was asked on MSE four years ago. That question quotes me as saying "Attempting to get around these set-theoretic issues leads you to model categories." Today, I'd stop short of that claim. Yes, model categories are one solution to the size problem, but they are not the only solution, and I don't think you're "naturally led" to them unless you set out to mimic the construction of the homotopy category of topological spaces.
A better motivation for why you'd want a model category $M$ is that it gives you control over the hom spaces in the homotopy category $Ho M$. Let $X,Y\in M$ and let $\gamma: M\to HoM$ the canonical functor. Let $Q$ denote cofibrant replacement, $R$ denote fibrant replacement, and $\sim$ denote the homotopy relation. Mark Hovey's book contains the Fundamental Theorem of Model Categories (Theorem 1.2.10), which states that $M(QX,RY)/\sim \cong Ho M(\gamma X, \gamma Y)$. Hence, if you have a model structure, it is much easier to compute hom sets in the homotopy category. Thanks to this theorem, if $M$ is locally small (i.e., has a set of morphisms between any two objects) then so is $HoM$, because we have just identified $Ho M(\gamma X, \gamma Y)$ as a quotient of a set.
To answer Maxime's comment, let me describe the context for my talk in 2012. This was back in the era when some folks were advocating to do away with model categories entirely and do everything $\infty$-categorically. It was before your time, but you might enjoy watching sparks fly at the MO thread "Do we still need model categories?" and other back-and-forth exchanges on many other threads in that era, between the same participants. My point in the first version of this answer was that the size motivation was not convincing to the audience because they responded with "hey, wait, $\infty$-categories also have locally small homotopy categories, so I guess we don't need model categories for this." I know that you, Maxime, know this, but let me spell it out for other readers. If $C$ is a quasi-category, then by definition it is a simplicial set satisfying a certain lifting condition. Its morphisms are the 1-simplices so it's automatically locally small. Its homotopy category is again locally small.
Another reason that the audience was not convinced by the size motivation is that one can also start out from a relative category or a category with weak equivalences $(C,W)$, which is much less structure than a model category requires, construct its homotopy category, then try to prove the homotopy category is locally small. One method of doing this is described here. The idea is to apply hammock localization to get to a simplicially enriched category, then apply $\pi_0$ to get to the homotopy category of $(C,W)$, and it'll often be locally small. A worthwhile paper to read is Function complexes in homotopical algebra by Dwyer and Kan. They start out by describing the hammock localization $L^H(C,W)$ for the situation when $C$ is a small category, but then they point out in 3.3 that they can do the same even when $C$ is not small, by way of a theory of homotopically small simplicial sets, meaning that $\pi_n(X)$ is small for all $n$, even if $X$ is not a small simplicial set. As far as I know, they only show that in the examples they care about, the simplicial category they get from $C$ is homotopically small, and do not provide general conditions on $C$ to guarantee this.
ASIDE: For the set-theoretically minded, a reasonable question is "Can quasi-categories actually model non-small settings?" or "What's the quasi-category associated to a non-small model category?" Usually folks in the $\infty$-category world get around these questions by way of Grothendieck universes. This means "small" now refers to a $U$-small set, and by fixing a sufficiently large universe $U$, formerly "large" categories become small categories. For example, Lennart Meier's paper Fibration Categories are Fibrant Relative Categories constructs a functor from the category of small relative categories to the category of (necessarily small) quasi-categories. Then he applies this functor to the relative category underlying a model category. What if the model category is not small? Meier has a footnote on page 7 stating that universes can get around this kind of objection. In Lurie's Higher Topos Theory the language of universes is similarly invoked. This use of universes might not be satisfying to the OP, because we could have applied that to the motivating problem in the first place and just used Gabriel-Zisman localization. Quillen would not have been satisfied with an answer of "just increase the size of $U$." The trick is to construct the homotopy category of a locally small category $C$, with a chosen class of morphisms $W$, and get a locally small homotopy category with respect to the same Grothendieck universe. As is pointed out in Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer, Kan, Hirschhorn, and Smith, "the homotopy category of a locally small homotopical category need not also be locally small." You really do need conditions on $(C,W)$ or you need to change the question being asked by increasing the size of the Grothendieck universe.
Anyway, coming back to the motivating question, a good example to share with the audience is Example 4.15 of Krause's "Localization theory for triangulated categories," a derived category that is not locally small. This shows that the issue can be subtle. Model categories provide an excellent way to get control over the hom sets of $HoM$, but $\infty$-categorical techniques can also be used to show that $HoC$ is locally small, for sufficiently nice $C$ (requiring less of $C$ than a full model structure). More details about why the homotopy category of a model category is locally small can be found in this nice blog post.
