I'm not sure I'm answering your question, but let me say some (vague and general) things and hopefully they'll be helpful.

Algebraic geometry over algebraically closed fields is really nice, essentially because of the Nullstellensatz: points are actually points, and functions are actually functions on these points. So it's often easy (or classical) to define a property of a variety over an algebraically closed field.

But then Grothendieck comes along and sez, hey, we should say things relatively, over an arbitrary base scheme. And the procedure for that is this: if you have a class of varieties C over algebraically closed fields, you extend it to the relative situation by saying that a map of schemes X --> S is in C if it is flat (which, experience and several nice theorems tell us, amounts to saying that the fibers are continuously parametrized by S) and each geometric fiber is in C.

There are two things that this buys you right off the bat: first, C is closed under base change, and second, C satisfies fpqc descent. Both are great indications that you have a good in-families notion; for instance they are certainly necessary if you want to make a good moduli functor out of C. If you didn't use geometric points, you couldn't guarantee fpqc descent, only Zariski descent.

But it seems like you were more interested in the converse: why should, if we have a good in-families notion, it be sufficient to check it on geometric fibers? Well actually, here "geometric" has nothing to do with it: you should always be able to check on fibers, because you want to be talking about a family of elements of C parametrized by the base. It's just that over non-algebraically closed fields it's probably harder to say what it means to lie in C -- if you don't do it geometrically, you potentially lose stability under base change and descent.