  As galaxy redshift surveys expand to larger areas on the sky, effects coming
from the curved nature of the sky become important, introducing wide-angle (WA)
corrections to the power spectrum multipoles at large galaxy-pair separations.
These corrections particularly impact the measurement of physical effects that
are predominantly detected on large scales, such the local primordial
non-Gaussianities. In this paper, we examine the validity of the perturbative
approach to modeling WA effects in the power spectrum multipoles for upcoming
surveys by comparing to measurements on simulated galaxy catalogs using the
Yamamoto estimator. We show that on the scales $k \lesssim 2\pi/\chi$, where
$\chi$ is the comoving distance to the galaxies, the estimated power spectrum
monopole differs by up to $5\%$ from the second-order perturbative result, with
similar absolute deviations for higher multipoles. To enable precision
comparison, we pioneer an improved treatment of the $\mu$-leakage effects in
the Yamamoto estimator. Additionally, we devise a solution to include $f_{\rm
NL}$ in the perturbative WA calculations, avoiding divergences in the original
framework through the integral constraint. This allows us to conclude that WA
effects can mimic a $f_{\rm NL}\sim5$ signal in the lowest SPHEREx redshift
bin. We recommend using non-perturbative methods to model large scale power
spectrum multipoles for $f_{\rm NL}$ measurements. A companion paper, Wen et
al. 2024, addresses this by introducing a new non-perturbative method going
through the spherical Fourier-Bessel basis.
