This question is a follow up to a previous question: Spherical Harmonic Identity.
Instead of using the above question's method, I tried something like this, but don't get the same result and I'm unsure why that is.
$$\partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)=\partial_{\theta_0} Y_{\ell,0}^*(\theta,\phi)Y_{\ell,0}(\theta_0,\phi_0)+\partial_{\theta_0} \sum_{m=-\infty}^{-1}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)+\partial_{\theta_0} \sum_{m=1}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)$$ Then switching the final term to a sum over negative $m$, I get: $$\partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)=\partial_{\theta_0} Y_{\ell,0}^*(\theta,\phi)Y_{\ell,0}(\theta_0,\phi_0)+\partial_{\theta_0} \sum_{m=-\infty}^{-1}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)+\partial_{\theta_0} \sum_{m=-\infty}^{-1}Y_{\ell,-m}^*(\theta,\phi)Y_{\ell,-m}(\theta_0,\phi_0).$$ The two sums can be joined. I also use the identity $Y_{\ell,-m}=(-1)^mY_{\ell,m}^*$. $$\partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)=\partial_{\theta_0} Y_{\ell,0}^*(\theta,\phi)Y_{\ell,0}(\theta_0,\phi_0)+\partial_{\theta_0} \sum_{m=-\infty}^{-1}(Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)+Y_{\ell,m}(\theta,\phi)Y_{\ell,m}^*(\theta_0,\phi_0)).$$ Expanding the final term to associated Legendre Polynomials, the result becomes $$\partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)=\partial_{\theta_0} Y_{\ell,0}^*(\theta,\phi)Y_{\ell,0}(\theta_0,\phi_0)+\partial_{\theta_0}\sum_{m=-\infty}^{-1}\left(\frac{2\ell+1}{4\pi}\right)\frac{(\ell-m)!}{(\ell+m)!}P_{\ell,m}(\cos\theta)P_{\ell,m}(\cos\theta_0)(e^{i(\phi-\phi_0)}+e^{-i(\phi-\phi_0)}).$$ Then, using the convention $$P_{\ell,m}=(1-x^2)^{m/2}\frac{d^m}{dx^m}P_\ell(x),$$ and the identity $$\frac{d}{dx}P_{\ell,m}(x)=\frac{mx}{(1-x^2)^{1/2}}P_{\ell,m}(x)-P_{\ell,m+1}(x),$$ the formula above becomes: $$\partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)=\partial_{\theta_0}Y_{\ell,0}^*(\theta,\phi)Y_{\ell,0}(\theta_0,\phi_0)+\sum_{m=-\infty}^{-1}\left(\frac{2\ell+1}{4\pi}\right)\frac{(\ell-m)!}{(\ell+m)!}P_{\ell,m}(\cos\theta)(m\cot\theta_0 P_{\ell,m}(\cos\theta_0)-P_{\ell,m+1}(\cos\theta_0))(2\cos(m\phi-m\phi_0)).$$ Finally, I set $\theta_0=0$, which reduces the above equation to: \begin{align} \partial_{\theta_0} \sum_{m=-\infty}^{\infty}Y_{\ell,m}^*(\theta,\phi)Y_{\ell,m}(\theta_0,\phi_0)&=\sum_{m=-\infty}^{-1}\left(\frac{2\ell+1}{4\pi}\right)\frac{(\ell-m)!}{(\ell+m)!}P_{\ell,m}(\cos\theta)\nonumber\\&(m\cot\theta_0 \delta_{m,0}-\delta_{m,-1})(2\cos(m\phi-m\phi_0))\nonumber\\ &=-2\left(\frac{2\ell+1}{4\pi}\right)\frac{(\ell+1)!}{(\ell-1)!}P_{\ell,-1}(\cos\theta)\cos(\phi-\phi_0)\nonumber\\ &=2\left(\frac{2\ell+1}{4\pi}\right)P_{\ell,1}(\cos\theta)\cos(\phi-\phi_0)\nonumber \end{align}
If there is an error here, I can't find it, maybe someone else can?
I think it might be a problem of the form of the associated Legendre function derivative... from the convention, it can't be mx/(1-x^2)^1/2, is it actually -mx/(1-x^2)? However if we replace x by costheta, and derive by theta, the formula seems the same.