What is the point of a local base?

Frequently, there are too many open sets. It is nice to be able to verify topological assertions (like continuity, for example) in terms of a smaller subclass which is more manageable.

As an analogy, consider the notion of basis of a vector space. It is quite good that we can, for instance, have a linear map entirely determined by its values on a basis. The concept of basis helps us to represent the whole structure in terms of a smaller, hopefully more manageable subclass. This is the idea.

Is it because the $B_x$ I described isn't the only local base?

Basically, yes. The point of talking about local bases is not to know that a local base exists: as you've observed, that is trivial. The point is to know that a nice local base exists. That is, we want to know that there exists a local base consisting of sets with certain nice properties, or a local base which itself has nice properties as a set (e.g., being countable), or a local base that is simply easy to think about. Typically the collection of all open sets containing $x$ won't be especialy nice, but there might be a smaller local base that is much nicer. And such a nicer local base $B_x$ can be used to understand the topology at $x$, since to test whether a set $U$ is a neighborhood of $x$ you just have to test whether $U$ contains some element of $B_x$.

If you look at all the applications you're referring to, none of them are talking about whether there exists an arbitrary local base. All of them are talking about whether there exists an local base that is nice in some special way. For instance, a space is first-countable if there exists a countable local base at every point. This has all sorts of powerful consequences: for instance, in a first-countable space a set $A$ is closed iff it is closed under taking limits of sequences.