Question: In most texts I have seen applying function $f$ to an element $x$ is denoted by $f(x)$, as opposed to $xf$ or $(x)f$ or $x(f)$. This makes function composition inconvenient. Given a diagram $A \xrightarrow{f} B \xrightarrow{g} C$ I need to write $f$ and $g$ in the opposite order of how I read them off from the diagram: $(g\circ f)(x)$ as opposed to $(x)(f\circ g)$.

**Question**: Is there a benefit to the current notation that I am missing? Or is this just unfortunate notation that happened to stick over the years?

Answer: 
It is perfectly possible to write functions on the right, like $xf$ or $x^f$, and this is commonly done in some areas, for example, with permutations in group theory. A consequence is that a product of cycles multiplies from left to right, exactly as you mention above, rather than right to left which can seem "backwards" (but it is done that way in many elementary textbooks).

However, for notation like $\sin x$ the "left side" notation is too-well established historically, and changing it to something like $x^{\sin{}}$ is probably not going to happen.

When the argument is written out, like the $x$ in $\sin(x)$, which side of the function you use doesn't matter too much, since the position of the $x$ directs the reader to perform the operations in the right order. It's only in a context where the argument is hidden, like with permutations, or in an arrow diagram like in your example, that the left-side convention can seem backwards. But you can (and I do like to) switch to right-side convention in such cases.
To add to @Ted's answer.

I think it has its origin in our usage (in Europe & the US) of language. We say "a function applied to", which suggest to write $f(x)$. However, for the same reason, we tend to read functional composition (as you pointed out) from left to right. So, it is a matter of what you want to put your focus on. If in your subject not much functional composition is involved, you can stay with the $f(x)$ notation. However, if you use functional composition a lot, like in permutation group theory, you might prefer $xf$ or $x^f$.

Let me point out that I have seen a lot of papers in group theory that actually use both conventions. For functions (and transformation) the right-action notation is used, but for "higher operators" that are never composed, the other notation is used. An example would be, if you associate another object (like a graph) with some given object $X$, people tend to write $G(X)$ while using the $xf$ notation for "ordinary" functions.

I do not speak Hebrew or Arabic, so I cannot tell how strong this habit might be for people that read from left to right.