= Solution
The surviving conclusion is a <quasi-isometry> from a <word metric> for a possibly infinite <generating set of a group>. Finite generation cannot be asserted. Choose $x\in X$ and $R\geq0$ so that $Gx$ is $R$-dense; the <cobounded group action> hypothesis provides such a choice. Put
$$
S=\{g\in G\setminus\{1\}:d_X(x,gx)\leq2R+1\}.
$$
This set is symmetric because $d_X(x,g^{-1}x)=d_X(gx,x)$. The <metric geodesic> subdivision argument in the next part gives
$$
\boxed{|g|_S\leq d_X(x,gx)+1,\qquad d_X(x,gx)\leq(2R+1)|g|_S.}
$$
In particular $S$ generates $G$. Apply the same estimates to $g^{-1}h$ to obtain the two-sided bounds for the <orbit map>; its image is $R$-dense. Thus \b[a cobounded isometric action on a <metric geodesic> space admits an orbit <quasi-isometry> for a suitable, possibly infinite, generating set]. This is the <cobounded orbit map lemma>.
Infinite stabilizers cause no obstruction to these bounds: all their nonidentity elements belong to $S$ and have length one, although their orbit displacement is zero. The additive constant permits precisely this collapse. For example, every group acts coboundedly on a point; its complete <Cayley graph> for $S=G\setminus\{1\}$ is bounded, but the group need not be finitely generated.
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