I've been thinking a little about irreducible components while reading Atiyah and Macdonald and wanted to try and understand the idea of irreducible component in more general terms of separation axioms. At one point, the book asks you to verify that the irreducible components of a Hausdorff space are individual points.
My question was: how strong is that requirement?
I think it is the same as $T_1$. If a space $X$ has irreducible components being its points, then for any $x,y\in X$ the subspace $\{x,y\}$ is not irreducible so there must be open subsets $U,V$ of the subspace which are nonempty and don't intersect and by definition of the subspace topology these must rise to open subsets $U', V' \subseteq X$ which are not necessarily disjoint but $x\in U', y\notin U'$ and vice versa for $V'$. Thus $X$ is $T_1$. A similar argument seems to go the other way.
So I have 2 main questions: is this correct and secondly, does this mean that quotienting out a space by the equivalence relation of $x\sim y$ iff $x,y$ are in the same irreducible component is the $T_1$-ization of a space, i.e. the maximal $T_1$ quotient?
First, you're not quite correct. If every irreducible component is a point, then you must be $T_1$ (fix two points $a \neq b$. Then the closure $\text{cl}(\{a,b\})$ cannot be irreducible, so we can find two closed sets $F$ and $G$ covering it, neither of which covers the whole set. Then the complements $F^c$ and $G^c$ show the $T_1$ separation of $a$ and $b$). However, you can be $T_1$ with nontrivial irreducible components. For example, the maximal irreducible component of $\mathbb{A}^2$ with the zariski topology is itself. Moreover, this space is $T_1$ and not a point.
However, there is a "$T_1$-ization" of a general space. Indeed, the category of $T_1$-topological spaces is reflective in the category of all topological spaces. Unwinding the category theoretic jargon, being a "reflective subcategory" means that there is a free way to turn an object of the big category into an object of the subcategory. For instance, abelian groups are reflective in groups, since there's a free way to abelianize a general group. For us, this tells us there's a free way to "$T_1$-ify" a topological space. (Indeed, many separation axioms give reflective subcategories of $\mathsf{Top}$).
Because of the subtle error in the first part of your question, your construction of the $T_1$-ization is also slightly flawed, but we can repair it! We can concretely describe the $T_1$-ification functor as the quotient of $X$ under some equivalence relation, described in detail here.
I hope this helps ^_^