If I understand your question correctly, you want to know why LAMMPS provides different minimisation algorithms? The reason is that different algorithms work better for different situations.
The default conjugate gradient style is typically a good balance between computational efficiency and stability, but there are probably situations in which it is better to use a more computationally intensive but more stable scheme, like a quasi-Newton method.
I suppose your first point it is true in theory, but in practice there might be subtle differences when dealing with truncated potentials.