This is a serious question despite provocative title. Ever since I found out about Cox's theorem, I got quite enthusiastic about an alternative approach to formalising probability theory and started thinking about what are the consequences of our standard measure-theoretic school.
This brings me to Borel–Cantelli lemma and its ability to show that a sequence of events will stop happening, almost surely, provided their probabilities are summable.
Nice theoretic result, but practitioners of applied probability stress the importance of actually controlling the rate of convergence. In other words, the fact that an event will stop happening doesn't tell us how far down the sequence we have to go for the result to be true: $1, 5, \dots, $ Graham' number.
So whilst I don't deny the validity of Borel-Cantelli, I merely want to know whether the emphasis on the significance of almost-sure convergence is misplaced and more mathematicians should strive to control the immediate rate of convergence rather than showing that the rate of converge is eventually dominated by a summable sequence.
Answer 1: Can you not say the same thing about a convergence of any sequence?
No, many proofs I've seen construct $N_{\varepsilon}$ for any $\varepsilon > 0$ in a proof of convergence, so the rate of convergence is not hard to extract. Also the concept of uniform convergence helps us to control the rate.
Answer 2: Pure mathematicians shouldn't worry about immediate applications of their results, so your question is misdirected.
Indeed, but what if there's a strong bias to care about a particular set of problems just as a consequence of axiomatising probability theory? Had we started with Cox's approach instead, would propositions like "In this sequence of events, the events will stop happening, eventually" be given as much attention?
Answer 3: Dealing with infinities often lead to counter-intuitive results, which often, in retrospect, are shown to be useful much later.
After many years since accepting measure-theory as underlying axioms, shouldn't we start looking at what kind of useful things came out of it and questioning whether our axioms were indeed conductive to useful research?
The mistake here is to say that $\mathcal S=\left\{ w : |\{n : S_n(w) > \sqrt{2 n \log \log n}\}| = \infty \right\}$ is at most countable. As a counterexample, consider the set $\mathcal S'$ of all real numbers in $(0\,,1)$ whose binary expansion is $0.0^{n(0)}1^{n(1)}0^{n(2)}1^{n(3)}0^{n(4)}...$ , in which a string of $n(0)$ $0$s is followed by $n(1)$ $1$s, followed by $n(2)$ $0$s, and so on, where $n(k)=2^{m(k)}$ ($k=0,1,...$) and $\left(m(0),m(1),...\right)$ is a strictly increasing sequence of natural numbers. Then $\mathcal S'\subset\mathcal S$, but $|\mathcal S'|=\mathfrak c$.