Essentially because the Tannakian theory gives in the unipotent case (and only in
that case) a reasonably sized answer with an easy motivic interpretation.

For the size you should be aware that already in the topological situation the
group scheme associated by Tannaka theory to the fundamental group of $\mathbb
P^1$ minus three points (i.e., the free group on two generators) is
*huge*. For one thing each irreducible representation gives rise to a
reductive quotient and there are continuous families (i.e., positive dimensional
varieties) of such representations. From this one can see that the group scheme
maps onto a product of reductive groups where the index set are the points of
some algebraic variety (see
What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
for the case of $1$-dimensional representations).

Added to this is the fact that most of the topological representations do not
have geometric origin and hence have no motivic interpretation. If one looks at
$\ell$-adic representations of the fundamental group over $\mathbb Q$ of $\mathbb
P^1$ minus three points (or of suitable germs if one wants a theory over
$\mathbb C$) and adds mixedness assumptions, then the Tannakian category should
have a motivic interpretation which also should be independent (in some suitable
sense) of $\ell$ and should be comparable to its cristalline equivalent. This
however all depends on the Langlands program and hence is currently beyond our
reach.

If one sticks to unipotent representations then essentially all these problems
disappear. A unipotent representation (over some field $k$ of characteristic
$0$) of the free group $F$ on two elements factors through a nilpotent quotient $\Gamma$ of
$F$ and such a nilpotent quotient has a Malcev completion, a unipotent algebraic
group $G$ over $\mathbb Q$ of dimension the rank of $\Gamma$, such that the
Tannakian category of unipotent representations of $\Gamma$ over $k$ is
equivalent to the category of $k$-representations of $G$. Passing to the limit
gives us a pro-unipotent algebraic group $G_\infty$ over $\mathbb Q$ whose category of
$k$-representations is equivalent to the category of unipotent
$k$-representations of $F$. Furthermore, the Lie algebra of $G_\infty$ has a
nice cohomological description; it is the free Lie algebra generated by
$H_1(X,\mathbb Q)$, where $X$ is $\mathbb P^1$ minus three points.

The motivic side of things now comes along very gracefully: For one of several
natural categories that has an appropriate $H_1(X)$ in it there is a
corresponding relative Tannakian description of unipotent families of objects
over $X$. As examples we have unipotent variations of rational Hodge structures,
geometrically unipotent $\mathbb Q_{\ell}$-adic sheaves over $\mathbb Q$ and
successive extensions of constant $F$-iso-crystals. In all these cases these
categories are described by representations of a pro-unipotent algebraic group
object in the appropriate base category (rational Hodge structure, $\mathbb
Q_{\ell}$-adic sheaves over $\mathbb Q$ and $F$-isocrystals) and in all the
cases its Lie algebra is the free Lie algebra on $H_1(X)$.