How do you estimate $\left( \sum_k 2^{2k} |P_k f(x)|^2 \right)^{1/2}$ by $\sum_k 2^{2k} |P_k f(x)|$? Don't the weight mess things up at low frequencies (i.e.\ if $|P_k f(x)| = δ_{k k_0}$ then the former is $2^{k_0}$ and the latter is $2^{2k_0}$)?

One way to generalize geodesics is replacing the Levi-Civita connection by any other (torsion-free if you want) connection on $TM$. There will also be a corresponding notion of exponential map. I suspect you might be able to extract a connection out of your family of maps $\sigma_p$ by pulling vector fields back along $\sigma_p$, differentiating at $0 \in T_ M$ and pushing forward along $\sigma_p$.

In step 5, when you write "finite" do you mean integral? And if so how do you see that?