I am trying to think of an example of a bounded linear functional on $L^p$ that is not obviously of the form $f \to \int fg $ for some $g \in L^q$. Analogously, maybe an example of a non-trivial operator that is reduced to a matrix in linear algebra would be the derivative operator on polynomials of degree $\leq n$ since the derivative operator isn't an example that came from elementary linear algebra itself.
I want to see an example of how duality adds value.
Take a bounded linear operator $T:L^2\to L^2$ and some $g$ in $L^2$. Then the functional $$ f\in L^2 \mapsto \langle T(f),g \rangle \in \mathbb C $$ is not obviously of the form you mention. That it turns out to be of that form is a deeply relevant fact of functional analysis underlying the definition of the adjoint of $T$.