A basic question about calculation under geodesic frame

Question

I have been recently reading some books related to Ricci flow, and I am confused when it comes to the evolution of Levi-Citita connection. That is: If $(M^n,g)$ Is a closed Riemannian manifold, and we have the Ricci flow equation $\frac{\partial g_{ij}}{\partial t}=-2Ric_{ij}$, then we have the evolution of the Christoffel symbol as follows: $$ \frac{\partial \Gamma_{ij}^k}{\partial t}=\frac{1}{2}g^{kl}(\nabla_ih_{jl}+\nabla_jh_{il}-\nabla_lh_{ij}), $$ where $h=\frac{\partial g}{\partial t}$, $\nabla_{i}=\nabla_{\partial_i}$. To prove this equation, we can calculate in geodesic frame for a fixed $p$ since it is a tensor, that is $$ \frac{\partial \Gamma_{ij}^k}{\partial t}=\frac{1}{2}\frac{\partial g^{kl}}{\partial t}(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij})+\frac{1}{2}g^{kl}(\partial_i(\frac{\partial g_{jl}}{\partial t})+\partial_j(\frac{\partial_{il}}{\partial t})-\partial_l(\frac{\partial g_{ij}}{\partial t})). $$ However, we now calculate in the geodesic frame, we have $\Gamma_{ij}^k(p)=0$, consequently we have $$\frac{\partial \Gamma_{ij}^k}{\partial t}=\frac{1}{2}g^{kl}(\nabla_ih_{jl}+\nabla_jh_{il}-\nabla_lh_{ij}), $$ Which finishes the proof. This is the standard proof. My question here is, at p we have $g^{kl}(p)=\delta^{kl}$, then the final formula should be $$ \frac{\partial \Gamma_{ij}^k}{\partial t}=\frac{1}{2}\delta^{kl}(\nabla_ih_{jl}+\nabla_jh_{il}-\nabla_lh_{ij}), $$ Where $\delta^{kl}=1$, if k=l, else 0. But this seems wrong. I am not clear why. I have read many books and they seem to ignore $g^{kl}(p)$, for example Bennett Chow and Dan Knopf’s book “The Ricci flow: an introduction.”