I don't have much time, but maybe the following can lead you to a solution. I'm sloppy too, writing $G$ both for the algebraic group (over some finite field $k$) and for the set of points $G(k)$.

This is semilar to a special case of a formula of Humphreys on Deligne-Lusztig characters.
("Deligne-Lusztig characters and principal indecomposable modules". J. Algebra 62 (1980), no. 2, 299--303.) The special case says that

$\sum_{w \in W} R_{T_w}(1) = |W| St_G$,

where $W$ is the Weyl group and $T_w$ the rational maximal torus defined by $w$ (by twisting a fixed rational maximal torus $T$), and $St_G$ is the Steinberg character.  In
the simplest case, $T$ is split, and then the assignment $w \mapsto T_w$ induces a bijection between conjugacy classes in the Weyl group and rational conjugacy classes
of rational maximal tori. This is like the multiplicites you found for $GL_3$.  Also
signs come up, since the sign of the "dimension" of a DL representation is related to the parity of the split rank of the
corresponding torus.

[Note that $R_T(1) = Ind_B^G(1)$ if $T$ is split and lies in the rational Borel $B$.]

(Humphreys proved it for $G$ simply connected, semisimple, split algebraic groups over finite fields and Jantzen (unpublished)
generalised it to arbitrary reductive groups over finite fields.)

EDIT: $R_T(\theta)$ is the virtual representation of DL defined by the rational maximal torus $T$ and the character $\theta$ of the finite group $T(k)$.

For $GL_2$ the formula boils down to $(St_G+1)+(St_G-1) = 2St_G$.