Here,
$$\left\|f\right\|_2=\left(\int_{-1}^1\left|f(t)\right|^2\,dt\right)^{1/2},$$
where the integral is a Riemann integral.
I am confused because $\{f_n\}$, where $f_n(t)=|t|^n$, is Cauchy with respect to $\|\cdot\|_2$ but converges pointwise to a discontinuous function. However, it converges to $f\equiv0$ with respect to $\|\cdot\|_2$.
Do we care about the fact that $\{f_n\}$ converges pointwise to a discontinuous function?
You are correct that the example you gave does not provide evidence that the space is not complete, because it is a Cauchy sequence that does have a limit in the space. By creating an example with similar behavior, but having a jump in the middle instead of only an endpoint, you can demonstrate that the space has a Cauchy sequence with no limit in the space.
E.g., $f_n(t) =\sqrt[n]{\max(0,t)}$. It is relevant in proving this sequence has no limit in the space to consider its pointwise convergence, because if it were convergent it would also have a subsequence converging pointwise a.e. to the limit function, whereas any function equal almost everywhere to the pointwise limit of $f_n$ is discontinuous at $0$. But you don't need to think in those terms, if you know by other means that limits with respect to this norm are unique up to equivalence a.e..