For the most part, functions are presumed to take unknown constants as inputs. Passing a random variable into such a function outputs a random variable: random in, random out. A notable exception to this is the function $Pr()$.
Now, given two variables $X$ and $X_{1}$ and the function $f(x)=Pr(X=x)$, due to the nature of $Pr()$ we have the counterintuitive fact that $f(X_{1})\ne Pr(X=X_{1})$. Is there a more sophisticated notation that doesn't give rise to this oddity?
I don't think there is. Functions measure quantities, and this quantities can be deterministic or random. You'll probably never get around the fact that the function $$f(X) = P(X=x \mbox{ and } X=y)$$ is not "random" if $x\neq y$. It is a constant function. Your question is interesting, but the current notation of functions don't tell us if such function has random or determined value. We can argue that deterministic functions have random outputs in a singlet, but that's exactly the point here. If there's hope to somewhat give a characterization of functions that have a truly random output, that set would not even be closed under simple operations (as the example suggests). I'd guess that the only way to tell is defining an operator that tell us if the output of a function is a random variable.