Why didn't $\Pr \left( A \rightarrow B \right)$ catch on? Students are conditioned to thinking in terms of IF-THEN statements even before high school, and courses offered at the university level often lead to the formalization of material implication. Material implication is a function that two Boolean values and returns/outputs a single Boolean value. One can use 'True' and 'False', but here I will use '1' and '0' respectively. Material implication can be written as a piecewise function (especially since I am not aware of a nice way to format truth tables here).
$$p \rightarrow q \triangleq \begin{cases} 1 & p = 1 \land q = 1 \\ 0 & p = 1 \land q = 0 \\ 1 & p = 0 \land q = 1 \\ 1 & p = 0 \land q = 0 \end{cases}$$
An indicator function of $ A \subseteq \Omega$ is a map $\mathbb{I}_A: \Omega \mapsto \{ 0, 1 \}$, and can be written explicitly in a piecewise fashion.
$$\mathbb{I}_A(\omega) \triangleq \begin{cases} 1 & \omega \in A \\ 0 & \omega \not\in A \end{cases}$$
We will interpret $\Pr(A)$ to be $\int_{\Omega} \mathbb{I}_A(\omega)dP$, and $\Pr(A \rightarrow B)$ to be $\int_{\Omega} \mathbb{I}_A(\omega) \rightarrow \mathbb{I}_B(\omega) dP$. Seeing an implication operator in the integrand is a novelty, but this implication on indicators can be re-written in terms of a new indicator. Thus it is an expression compatible with our existing understanding.
$$\mathbb{I}_A(\omega) \rightarrow \mathbb{I}_B(\omega) = \mathbb{I}_{C}(\omega) \triangleq \begin{cases} 1 & \omega \in A \land \omega \in B \\ 0 & \omega \in A \land \omega \not\in B \\ 1 & \omega \not\in A \land \omega \in B \\ 1 & \omega \not\in A \land \omega \not\in B \end{cases}$$
$$\Pr(C) = \int_{\Omega} \mathbb{I}_C(\omega) dP$$
At some point students encounter the mathematical fact that $\Pr \left( A \rightarrow B \right) \neq \Pr \left( B | A \right)$. Some mathematicians were motivated to develop conditional event algebras that attempt to define these as equal, and I've asked for applications of such algebras in the past. The Goodman–Nguyen–Van Fraasen algebra is an example of a conditional event algebra.
But even without getting into conditional event algebras, $\Pr \left( A \rightarrow B \right)$ is a defined quantity in the standard Kolmogorov treatment of probability that I have yet to see used. Why are we not using it?

I think the answer's already in your question. It's fairly natural to use the notation of the sentential calculus within probability statements when what we'd be happy to take as the probability of the sentence's being true corresponds to what you get from totting up the probabilities for all the possible states of affairs for which the sentence is true. And so you sometimes see $\Pr(\lnot p)$, $\Pr(p \lor q)$, & the like, where $p$ & $q$ are sentences; more often $\Pr(\lnot A)$, $\Pr(A \lor B)$, & the like, where $A$ & $B$ are sets†. Material implication is exceptional, however: $\Pr(p \rightarrow q)$ ought to mean $\Pr(q|p)$, we feel—the probability of "if there are dark clouds now, then it will rain soon" depends only on how often it tends to rain soon given that there are dark clouds now—but that conditional probability isn't what we end up with by treating $\Pr(p \rightarrow q)$ in the same way as the others. If it hasn't caught on in standard Probability Theory, it's some combination of its being unintuitive as explained above, its already being "taken" by conditional events algebras, & lack of demand for an especially concise notation for what it represents in standard P.T.

† The latter usage is a generally harmless abuse of notation, but I wonder if it doesn't contribute to puzzlement about what $\Pr(A \rightarrow B)$ means.
