= Solution
Use radiation-era $H=C_*T^2/m_{\rm pl}$, with $C_*=1.66\sqrt{g_*}$. The stipulated interaction rate is $\alpha_Y^4T^5/m_Y^4$, not $\alpha_Y^2T^5/m_Y^4$. Decoupling therefore gives
$$
T_D^3=\frac{C_*m_Y^4}{\alpha_Y^4m_{\rm pl}},\quad\boxed{m_{Y\rm nr}=\frac{\alpha_Y^4m_{\rm pl}}{C_*}\simeq7\times10^5\ {\rm GeV}},\quad T_D=m_Y\left(\frac{m_Y}{m_{Y\rm nr}}\right)^{1/3}.
$$
At the threshold $T_D=m_{Y\rm nr}$, the assumed $g_*\simeq100$ is self-consistent with the stated high-temperature condition. Smaller masses decouple nonrelativistically, with $x_D=m_Y/T_D=(m_{Y\rm nr}/m_Y)^{1/3}\gg1$.
For an <thermal equilibrium> species with zero <chemical potential>, $n_Y=2(m_YT_D/2\pi)^{3/2}e^{-x_D}$, while $s=2\pi^2g_{*s}T_D^3/45$. Thus
$$
\frac{n_Y}{s}=\frac{45g_Y}{2\pi^2g_{*s}(2\pi)^{3/2}}x_D^{3/2}e^{-x_D}\simeq\boxed{10^{-3}\left(\frac{m_{Y\rm nr}}{m_Y}\right)^{1/2}e^{-(m_{Y\rm nr}/m_Y)^{1/3}}}.
$$
The numerical coefficient before rounding is about $2.9\times10^{-3}$ for $g_Y=2$ and $g_{*s}=100$. This is <mass-dependent fermion freeze-out>. It assumes the rate maintains <chemical equilibrium> until sudden <cosmological particle freeze-out> and that entropy is conserved afterwards. Elastic scattering alone would not justify the <thermal equilibrium> number-density estimate. For decoupling below the high-temperature range, changes in effective degrees of freedom modify the numerical estimate; the requested formula treats them as roughly constant.
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