  Scalar-Gauss-Bonnet (sGB) gravity with an additional coupling between the
scalar field and the Ricci scalar exhibits very interesting properties related
to black hole stability, evasion of binary pulsar constraints, and general
relativity as a late-time cosmology attractor. Furthermore, it was demonstrated
that a spherically symmetric collapse is well-posed for a wide range of
parameters. In the present paper we examine further the well-posedness through
$3+1$ evolution of static and rotating black holes. We show that the evolution
is indeed hyperbolic if the weak coupling condition is not severely violated.
The loss of hyperbolicity is caused by the gravitational sector of the physical
modes, thus it is not an artifact of the gauge choice. We further seek to
compare the Ricci-coupled sGB theory against the standard sGB gravity with
additional terms in the Gauss-Bonnet coupling. We find strong similarities in
terms of well-posedness, but we also point out important differences in the
stationary solutions. As a byproduct, we show strong indications that
stationary near-extremal scalarized black holes exist within the Ricci-coupled
sGB theory, where the scalar field is sourced by the spacetime curvature rather
than the black hole spin.
