How is Grover's Algorithm useful?
The thing you're missing is that you aren't search for $|3\rangle$, you're searching for "the input that makes $f$ return true" where $f$ is some function.

For example, maybe $f$ is a discrete logarithm checker like $x \rightarrow 2^x \equiv^? 432821 \pmod{7438927891}$. Or $f$ could interpret its input as a claimed proof of the Riemann hypothesis, and check if that proof is correct. Anything you can imagine checking, $f$ could be that.

So Grover's algorithm is very very widely applicable. But unfortunately it also only gives a quadratic advantage, so "applicable" often doesn't translate into "useful". For example, the space of (reasonably sized) possible Riemann-hypothesis proofs is so huge that it laughs at square roots.