  We explore the effects on nuclear bulk properties of using regularization
cutoffs larger than the nucleon mass within the chiral effective field theory
with a power counting that ensures order-by-order renormalization in the
two-nucleon system. To do so we calculate ground-state properties of the
$^{16}$O nucleus in the Hartree--Fock approach in a basis made up of plane
waves confined in a cube. We find that regularization cutoff effects manifest
themselves in two distinct ways: a strong sensitivity to the the counter-terms
in attractive singular partial waves (related to the sign of the corresponding
low-energy constant) and to the correction for spurious deeply bound states
(for high enough cutoffs). In fact the latter happens to deprive the
Hartree--Fock approximation of yielding bound solutions in nuclei. We conclude
that, when using a leading-order chiral potential in the Nogga--Timmermans--van
Kolck's power counting (with a regularization cutoff higher than the nucleon
mass), one cannot produce a selfconsistent mean field free of spurious
bound-state effects that can serve as a reference state for beyond-mean-field
methods. For high regularization cutoffs which yield an attractive $^3S_1$
contact potential, one can at best incorporate in the mean-field solution a
partial correction for spurious bound states. Then the remaining correction has
to be added to the residual interaction in a treatment beyond the Hartree--Fock
approximation. In fact a ``full'' correction in the $^3D_2$ channel, with
energy shifts of the order of or somewhat larger than those recommended in
Phys. Rev. C 103, 054304 (2021), is possible.
