Let $\sim$ be some symmetric and transitive relation on a set $X$. Couldn't the notion of a quotient set still be defined, where some elements may "vanish"?
For example, let $A=\{1,2,...,7\}$, and let $1\sim2\sim3$ and $5\sim6\sim7$. Could I define $A/\sim$ as $\{[1],[5]\}$, where the $4$ "vanishes"?
My thinking is that the only "purpose" of reflexivity is to guarantee the existence of an equivalence class for every element. If my idea has some merit, then is there a name for this kind of relation?
These are called partial equivalence relations (PERs). They come up frequently in the semantics of programming languages and related topics. The quotient-like operation you mention is also used, and works the way that you expect.
For instance, the PERs on Kleene's first algebra make up a notable subcategory of the effective topos, equivalent to the associated category of modest sets. These are the sets that are able to be represented by values in the algebra, such that each element of the algebra represents at most one element of the set (so, the sets cannot be bigger than the set of values in the algebra). The PER determines which values in the algebra represent equivalent elements, and which do not represent any element at all (the ones for which reflexivity fails). The quotient-like operation gives you (something isomorphic to) the represented set.