Why are numerical solutions necessary in order to plot this?
They aren't. Numerical solutions of the TDSE are only necessary when no analytical solution is known.
The only real-world reason for implementing a numerical solution of the Schrödinger equation when there are analytical solutions available that makes sense is when you're implementing a multi-purpose solver that needs to work robustly and consistently over a range of situations that includes potentials for which no analytical solution is known; it might then be impractical to switch back-and-forth from numerical to analytical solutions. (This is the case for the solver you've linked in the comments.)
Of course, one might also choose to solve numerically for situations where analytical solutions are known, as training for writing numerical solvers, or as testing for one of those, but those are not "necessary" in any sense.
If you have examples in front of you that don't seem to fit either of these patterns, it is on you to produce those examples.