  We address the unphysical energy dependence of quarkonium-hadroproduction
cross sections at Next-to-Leading Order (NLO) in alpha_S which we attribute to
an over-subtraction in the factorisation of the collinear singularities inside
the PDFs in the MSbar scheme. Such over- or under-subtractions have a limited
phenomenological relevance in most of the scattering processes in particle
physics. On the contrary, it is particularly harmful for P_T-integrated
charmonium hadroproduction which renders a wide class of NLO results
essentially unusable. Indeed, in such processes, alphaS is not so small, the
PDFs are not evolved much and can be rather flat for the corresponding momentum
fractions and, finally, some process-dependent NLO pieces are either too small
or too large. We propose a scale-fixing criterion which avoids such an
over-subtraction. We demonstrate its efficiency for eta(c,b) but also for a
fictitious light elementary scalar boson. Having provided stable NLO
predictions for eta(c,b) P_T-integrated cross sections, sigma^NLO(eta(Q)), and
discussed the options to study eta(b) hadroproduction, we argue that their
measurement at the LHC can help better determine the gluon PDF at low scales
and tell whether the local minimum in conventional NLO gluon PDFs around
x=0.001 at scales below 2 GeV is physical or not.
