" so why must the wave function for a hydrogen atom be an eigenfunction for 𝐿̂^2 and 𝐿̂_𝑧?" They must not, but since the H, L^2, L_z do commute with each other, they can have some mutual eigenfunctions.You do not have to choose these specific mutual eigenfunctions, you can choose some others, but I think it is obviously very convenient to work with such functions that are simultaneously eigenfunctions of all these three operators(and this is why people usually choose these ones).

" so why must the wave function for a hydrogen atom be an eigenfunction for $\hat{L}^2$ and $\hat{L}_z$?" They must not, but since the $\hat{H}$, $\hat{L}^2$, $\hat{L}_z$ do commute with each other, they can have some mutual eigenfunctions.You do not have to choose these specific mutual eigenfunctions, you can choose some others, but I think it is obviously very convenient to work with such functions that are simultaneously eigenfunctions of all these three operators(and this is why people usually choose these ones).