Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta_{p^n})$ (where $\zeta_{p^n}$ is a primitive root of unity of exact order $p^n$) for sufficiently large $n$. (See, e.g., Theorem 2 of these notes.) More generally, one gets a similar result for arbitrary $\mathbb{Z}_p$ extensions of number fields. Mazur and Mazur and Rubin have studied the variation of the $p$ part of the Tate-Shafarevich group of an elliptic curve in $\mathbb{Z}_p$ extensions and the variation of the rank of an elliptic curve in $\mathbb{Z}_p$ extensions.

I find this somehow unsatisfying. The restriction to the study of the $p$ part of the ideal class group & Tate Shafarevich, and the restriction to the study of $\mathbb{Z}_p$ extensions seem quite strong.

Yet I've gotten the impression that Iwasawa theory is considered to be fundamental in number theory. So I feel as though I'm missing perspective on why Iwaswa theory is important.

What are some important applications of Iwasawa theory?

I'd also be happy with high-level philosophical comments.