Noether's theorem has nothing to do with Buckingham's theorem.

Both of them are relevant to the project of exploiting symmetry, and deal with solidifying a mostly understood situation. (You can't apply Buckingham's theorem unless you know all the relevant pieces.) The more practical piece that I had in mind is that someone eager to apply Noether's theorem will need to look for symmetries; they may have found techniques for hunting for symmetries that will be useful in general. It might be worth looking into material that teaches it, not because it itself is directly useful, but because the community that knows it may know other useful things.

In what sense do you mean Buckingham's theorem is vacuous?

It's a quite bit more general than Lagrangian mechanics. You can extend it to any functional that takes functions between two manifolds to complex numbers.