  In this article we present an alternative formalism for the inflationary
phenomenology of rescaled Einstein-Gauss-Bonnet models which are in agreement
with the GW170817 event. By constraining the propagation velocity of primordial
tensor perturbations, an approximate form for the time derivative of the scalar
field coupled to the Gauss-Bonnet density is extracted. In turn, the overall
degrees of freedom decrease and similar to the case of the canonical scalar
field, only one scalar function needs to be designated, while the other is
extracted from the continuity equation of the scalar field. We showcase
explicitly that the slow-roll indices can be written in a closed form as
functions of three dimensionless parameters, namely
$x=\frac{1}{2\alpha}\bigg(\frac{\kappa\xi'}{\xi''}\bigg)^2$, $\beta=8H^2\xi''$
and $\gamma=\frac{\xi'\xi'''}{\xi''^2}$ and in turn, we prove that the
Einstein-Gauss-Bonnet model can in fact produce a blue-tilted tensor spectral
index if the condition $\beta\geq1$ is satisfied, which is possible only for
Einstein-Gauss-Bonnet models with $\xi''(\phi_k)>0$. Afterwards, a brief
comment on the running of the spectral indices is made where it is shown that
$a_{\mathcal{S}}(k_*)$ and $a_{\mathcal{T}}(k_*)$ in the constrained case are
approximately of the order $\mathcal{O}(10^{-3})$, if not smaller. Last but not
least, we examine the conditions under which the Swampland criteria are
satisfied. We connect the tracking condition related to scalar field theories
with the present models, and we highlight the important feature of the models
we propose that the tracking condition can be satisfied only if the Swampland
criteria are simultaneously satisfied, however the cases with $\xi\sim1/V$ and
$\xi\sim V$ are excluded, as they cannot describe the inflationary era
properly.
