  Precision measurements of the semileptonic decays $D_s^+ \to \eta e^+ \nu_e$
and $D_s^+ \to \eta^\prime e^+ \nu_e$ are performed with 7.33\,fb$^{-1}$ of
$e^+e^-$ collision data collected at center-of-mass energies between 4.128 and
4.226 GeV with the BESIII detector. The branching fractions obtained are
$\mathcal{B}(D_s^+ \to \eta e^{+} \nu_e)$ = $(2.255\pm0.039_{\rm stat}\pm
0.051_{\rm syst})\%$ and $\mathcal{B}(D_s^+ \to \eta^{\prime} e^{+} \nu_e)$ =
$(0.810\pm0.038_{\rm stat}\pm 0.024_{\rm syst})\%$. Combining these results
with the $\mathcal{B}(D^+\to\eta e^+ \nu_e)$ and $\mathcal{B}(D^+\to\eta^\prime
e^+ \nu_e)$ obtained from previous BESIII measurements, the $\eta-\eta^\prime$
mixing angle in the quark flavor basis is determined to be $\phi_{\rm P} =
(40.0\pm2.0_{\rm stat}\pm0.6_{\rm syst})^\circ$. Moreover, from the fits to the
partial decay rates of $D_s^+ \to \eta e^+ \nu_e$ and $D_s^+ \to \eta^\prime
e^+ \nu_e$, the products of the hadronic transition form factors
$f_+^{\eta^{(\prime)}}(0)$ and the modulus of the $c\to s$
Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ are determined by using
different hadronic transition form factor parametrizations. Based on the
two-parameter series expansion, the products $f^\eta_+(0)|V_{cs}| =
0.4519\pm0.0071_{\rm stat}\pm0.0065_{\rm syst}$ and
$f^{\eta^\prime}_+(0)|V_{cs}| = 0.525\pm0.024_{\rm stat}\pm0.009_{\rm syst}$
are extracted. All results determined in this work supersede those measured in
the previous BESIII analyses based on the 3.19 fb$^{-1}$ subsample of data at
4.178 GeV.
