(Edit: I focus on the case where $\underset{a}\lim f=\underset{a}\lim g=0$ (ie $f(a)=f(g)=0$ by extending $f$ and $g$ by continuity). But the other cases can be reduced to this case by considering $1/f$ and $1/g$)

It is worthwhile to remark that one of the only cases where the subtlety of L'Hopital's rule is actually needed is when $f$ or $g$ has a vertical tangent at $a$ (ie $f'$ or $g'\underset{a}{\rightarrow}±\infty$), which is rare. For instance : $\lim_{x\rightarrow1^-}\frac{\arccos(x)}{\sqrt{1-x}}$

In (almost) all other cases, when $f'(a)$ and $g'(a)$ exist and $g'(a)≠0$, the result is a mere consequence of the very definition of differentiability (namely, $f$ is differentiable at $a$ ⟺ $f(x)-f(a)\underset{x\rightarrow a}\sim (x-a)f'(a)$ ), since

$\frac{f(x)}{g(x)}=\frac{f(x)-\overbrace{f(a)}^{=0}}{g(x)-\underbrace{g(a)}_{=0}}\underset{x\rightarrow a}\sim\frac{(x-a)f'(a)}{(x-a)g'(a)}\underset{x\rightarrow a}\sim\frac{f'(a)}{g'(a)}$

(Edit: Even when $g'(a)=0$, it seems to me that L'Hopital's rule is often superfluous, because when $f$ and $g$ are differentiable enough times at $a$, one can apply Taylor's theorem to them, which is often faster than L'Hopital's rule which may require repeated derivatives.)

The beauty and subtlety of L'Hopital's rule stem from the fact that it doesn't require $f$ nor $g$ to be differentiable at $a$. Using this subtle and complex theorem in cases that are easily solvable with stronger assumptions and simpler properties is pedagogically absurd.

Edit: So, from my point of view, this is what explains the bad image of L'Hopital's rule: magical and often unproven so of little pedagogical value, and moreover quite useless compared to more efficient and easier to prove methods, except in some rare cases.

(It seems to me indeed that L'Hopital's theorem is especially popular in the U.S. curriculum compared to the rest of the world. For example, in France, it is totally absent from the curriculum, and is only marginally treated as an exercise. I find this much more consistent.)