Why does the Cauchy-Schwarz Inequality even have a name? When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof.

I've always thought the geometric definition of dot multiplication: 
$$|{\bf a }||{\bf b }|\cos \theta$$ is equivalent to the other, algebraic definition: $$a_1\cdot b_1+a_2\cdot b_2+\cdots+a_n\cdot b_n$$
And since the inequality is directly implied by the geometric definition (the fact that $\cos(\theta)$ is $1$ only when $\bf a$ and $\bf b$ are collinear), then shouldn't the Cauchy-Schwarz inequality be the world's most obvious and almost-no-proof-needed thing?

Can someone correct me on where my thought process went wrong?
Side note: it's actually the Cauchy-Schwarz-Bunyakovsky inequality, and don't let anyone tell you otherwise.

The problem with using the geometric definition is that you have to define what an angle is. Sure, in three dimensional space, you have pretty clear ideas about what an angle is, but what do you take as $\theta$ in your equation when $i$ and $j$ are $10$ dimensional vectors? Or infinitely-dimensional vectors? What if $i$ and $j$ are polynomials?

The Cauchy-Schwarz inequality tells you that anytime you have a vector space and an inner product defined on it, you can be sure that for any two vectors $u,v$ in your space, it is true that $\left|\langle u,v\rangle\right| \leq \|u\|\|v\|$.

Not all vector spaces are simple $\mathbb R^n$ businesses, either. You have the vector space of all continuous functions on $[0,1]$, for example. You can define the inner product as

$$\langle f,g\rangle=\int_0^1 f(x)g(x)dx$$

and use Cauchy-Schwarz to prove that for any pair $f,g$, you have
$$\left|\int_{0}^1f(x)g(x)dx\right| \leq \sqrt{\int_0^1 f^2(x)dx\int_0^1g^2(x)dx}$$

which is not a trivial inequality.
Flip your argument around. The Cauchy-Schwarz inequality makes thinking about angles seem outmodish, vestigial, obsolete. You don't need protractors to do geometric analysis anymore. You just need algebra: just one quadratic inequality. So much of what your proof using $\cos \theta$ has been doing only requires a simple algebraic inequality, and neither $\cos$ nor $\theta$.

Why algebra is important is out of scope to this answer.