ag.algebraic geometry - Why does the algebraic condition of flatness on the structure sheaves give a good definition of family?

There is also the following (probably unhistorical) point of view (it is a version of Hailong Dao's answer). Namely, you don't have to work with flat families at all, so if you want, you can just declare all morphisms to be families'. The problem with this approach is that
this is a family ofderived' objects. Here's an example:

Let $S$ be a scheme, and let $F$ be a coherent sheaf on S. When is it a `family' of its fibers? If it is flat, it definitely deserves to be called a family of vector spaces (a vector bundle). But even if it is not flat, you can still view it as a family, but the family of what? The (derived) fibers of $F$ are no longer vector spaces, they are complexes of vector spaces (precisely because $F$ fails to be flat), so we can view $F$ as a nice family of complexes of vector spaces, even though $F$ itself is a sheaf, not a complex.

To summarize: by all means, let's forget about flatness and declare any morphism to be a family... of some kind of derived objects. If we now want members of the family to be actual objects (schemes, vector spaces, sheaves, or whatever it is we are trying to include in a family), flatness is forced on you more or less by definition.