  The Dark Energy Spectroscopic Instrument (DESI) will provide unprecedented
information about the large-scale structure of our Universe. In this work, we
study the robustness of the theoretical modelling of the power spectrum of
FOLPS, a novel effective field theory-based package for evaluating the redshift
space power spectrum in the presence of massive neutrinos. We perform this
validation by fitting the AbacusSummit high-accuracy $N$-body simulations for
Luminous Red Galaxies, Emission Line Galaxies and Quasar tracers, calibrated to
describe DESI observations. We quantify the potential systematic error budget
of FOLPS, finding that the modelling errors are fully sub-dominant for the DESI
statistical precision within the studied range of scales. Additionally, we
study two complementary approaches to fit and analyse the power spectrum data,
one based on direct Full-Modelling fits and the other on the ShapeFit
compression variables, both resulting in very good agreement in precision and
accuracy. In each of these approaches, we study a set of potential systematic
errors induced by several assumptions, such as the choice of template
cosmology, the effect of prior choice in the nuisance parameters of the model,
or the range of scales used in the analysis. Furthermore, we show how opening
up the parameter space beyond the vanilla $\Lambda$CDM model affects the DESI
observables. These studies include the addition of massive neutrinos, spatial
curvature, and dark energy equation of state. We also examine how relaxing the
usual Cosmic Microwave Background and Big Bang Nucleosynthesis priors on the
primordial spectral index and the baryonic matter abundance, respectively,
impacts the inference on the rest of the parameters of interest. This paper
pathways towards performing a robust and reliable analysis of the shape of the
power spectrum of DESI galaxy and quasar clustering using FOLPS.
