Is something wrong with my luminosity calculation?
Yes it would.
It is that way because the effective temperature is defined to be $(L/4\pi \sigma R^2)^{0.25}$.
The radius of a neutron star is about 10 km $(1.4\times 10^{-5}R_\odot)$. They are born with surface temperatures of around $10^8$ K. The coldest white dwarfs have effective temperatures of about 3000 K.
The luminosity ratio is
$$ \frac{L_{\rm NS}}{L_{\rm WD}}  = \left(\frac{10^{8}}{3000}\right)^4 \left(\frac{1.4\times 10^{-5}}{0.01}\right)^2 = 2.4\times 10^{12}\ .$$
Thus the most luminous neutron stars are (briefly) much more luminous than the coldest white dwarfs. They would remain more luminous until they cooled to $\sim 10^5$K, which could take as long as $10^7$ years (Lattimer &amp; Prakash 2004).