  We present novel improvements in the context of symbol-based multigrid
procedures for solving large block structured linear systems. We study the
application of an aggregation-based grid transfer operator that transforms the
symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the
coarser level. Our convergence analysis of the Two-Grid Method (TGM) reveals
the connection between the features of the scalar-valued symbol at the coarser
level and the properties of the original matrix-valued one. This allows us to
prove the convergence of a V-cycle multigrid with standard grid transfer
operators for scalar Toeplitz systems at the coarser levels. Consequently, we
extend the class of suitable smoothers for block Toeplitz matrices, focusing on
the efficiency of block strategies, particularly the relaxed block Jacobi
method. General conditions on smoothing parameters are derived, with emphasis
on practical applications where these parameters can be calculated with
negligible computational cost. We test the proposed strategies on linear
systems stemming from the discretization of differential problems with
$\mathbb{Q}_{d} $ Lagrangian FEM or B-spline with non-maximal regularity. The
numerical results show in both cases computational advantages compared to
existing methods for block structured linear systems.
