It's often nice to think of natural numbers as a subset of the integers. And to think of the integers as a subset of the rationals. The rationals as a subset of the algebraic. The algebraic as a subset of the reals.
And some people even like to think of the reals as a subset of the complex.
But there are lots of things going on there since those numbers are all used as objects and as operations. But keep in mind that it's nice to have a playground where all your toys are already there.
Now let's bring in the physics. In physics you don't need to use $\mathbb R^3$ technically you could use anything bijective with that as a model. And same for spacetime, you don't have to use $\mathbb R^4$ because anything bijective could work as well. As a vector space you could map over the vector addition and the scaling by scalars. But if you want to respect a metric that wouldn't be enough.
When you have a metric you can also talk about hyperplanes, and the vectors orthogonal to them. You might want to talk about the plane spanned by two vectors. You might want to talk about the 3d subspace spanned by three vectors. You might want to talk about the reflection across a hyperplane orthogonal to a vector.
So lots of objects and lots of operations. And just like you liked having a larger algebra that contained your smaller algebras, so you might might want an algebra everything can live inside.
A mathematician will say something isn't a vector if you can't add it. And even if you can add then, they would say it's just a group if you can't also scale it by a scalar. But the vectors are just a subset of the full multivector algebra. And they aren't even a subalgebra. So they aren't really a finished system. The only way you could they are finished is if you pretend you can't multiply them.
By that standard the imaginary numbers are just a vector space because you can pretend not to be able to multiply them.
But in physics we have things like spin and angular momentum and magnetic fields that truly require planes to describe them. And in relativity you have to admit planes as new objects because in 4d a 2d plane isn't a hyperplane so you can't just point at some vectors and give them a different name.
So we need scalars and we need vectors and we need planes too. That's life, and geometry.
If we introduce all these objects we can then talk about their relationships. But each one is sometimes an object and sometimes an operator. Just like matrices. A square matrix can multiply another square matrix treating the other like an object when that matrix could itself be an operator.
That's what an algebra is fundamentally.
So you want to have your objects and your operations. In fact most objects could just be transformations of reference objects. And the quadratic form is just trying to specify the metric to be respected by the transformations, the operations. Which end up giving you the full algebra.
So the set of products of vectors is natural as a set of operations. The full algebra isn't exactly needed, but it can be useful as a basis.