Why do we care about specifying events in a probability space? Why aren't probability spaces just defined as $(\Omega, p)$ pairs with $\Omega$ as the sample space, $\sum_{\omega \in \Omega}p(\omega) = 1$, and for a subset $A \subseteq \Omega$, $\Pr(A) := \sum_{\omega \in A}p(\omega)$? Said another way, why aren't all $(\Omega, \mathcal{A}, p)$ probability spaces of the form $(\Omega, \mathcal{P}(\Omega), p)$? What do we gain by giving ourselves the freedom to exclude certain subsets of $\Omega$ from $\mathcal{A}$ ?

That's a good question. An answer is that there are many probability spaces $(\Omega,\mathcal{A},p)$ where the probability function $p$ cannot be extended to all of $\mathcal{P}(\Omega)$. For example, consider the probability space where $\Omega=[0,1]$, $\mathcal{A}$ is the Lebesgue $\sigma$-algebra on $[0,1]$, and $p=\lambda$ is the Lebesgue measure. Then there is no way of extending $p$ to have a value when given the Vitali set (the standard example of a non-Lebesgue measurable subset of $[0,1]$).