Hopf Umlaufsatz (circulation/rotation index/rotation angle theorem) on a Riemannian 2-manifold

Question

The book I am studying (Loring W. Tu's Differential geometry, Connections, Curvature and Characteristic Classes) makes use of the following theorem:

Theorem 17.4 (Hopf Umlaufsatz). Let $(U,e_1,e_2)$ be a framed open set on an oriented Riemannian 2-manifold, and $\gamma: [a,b] \to U$ a positively oriented piecewise smooth simple closed curve. Then the total change in the angle of $T(s)=\gamma(s)$ around $\gamma$ is $$\sum_{i=1}^{m}\Delta\zeta_i+\sum_{i=1}^{m}\epsilon_i = 2\pi$$

where $\epsilon_i$ has been previously defined as the jump angle of $T$ at the vertex $i$ of the curve and $\Delta\zeta_i$ the change of the angle along the edge $\gamma([s_(i-1),s_i])$. The angle function has been previously defined with respect to a positively oriented frame and the curve has $m$ vertices.

All the proofs of the theorem I have found, including the one provided in the book, prove Hopf Umlaufsatz not for a framed open set of an oriented Riemannian 2-manifold, but for a simple curve in the plane ($\mathbb{R}^2$). Whilst this makes sense intuitively, I wanted to check rigorously all the passages as I had the following proof in mind:

Given the frame $(e_1,e_2)$ of the set (call it $S \subset M$) we use a map $\phi$ with coordinates $(x^1,x^2)$ $^{(1)}$, such that $e_i=\frac{\partial}{\partial{x^i}}$. The curve $\gamma: [a,b] \to U$ maps on the curve $\phi(\gamma): [a,b] \to \mathbb{R}^2$ which is simple as well $^{(2)}$. The angle functions of $\gamma$ and $\phi(\gamma))$ coincide $^{(3)}$, i.e. $\zeta(\gamma(t))=\zeta(\phi(\gamma(t)))$ as well as the jump angles $\epsilon_i$ so we have $\sum_{i=1}^{m}\Delta\zeta_i(\phi \circ \gamma)+\sum_{i=1}^{m}\epsilon_i(\phi \circ \gamma)=\sum_{i=1}^{m}\Delta\zeta_i(\gamma)+\sum_{i=1}^{m}\epsilon_i(\gamma) = 2\pi$

I am pretty confident about 2 and 3, but I am not sure about 1:

1. The frame $e_1,e_2$ is a coordinate frame iff $[e_1,e_2]=0$ (see here for instance). Now, we don't have the guarantee that the frame $(e_1,e_2)$ is commuting, but can we choose $\widetilde{e_1}$ and $\widetilde{e_2}$ from $e_1,e_2$ such that ($\widetilde{e_1}$ , $\widetilde{e_2}$) is a commuting frame?
2. Because $\phi$ is a diffeomorphism, hence bijective
3. By definition, as we have, in local coordinates, that $T=\gamma^{'}=\frac{d}{dt}x^1(\gamma(t))\frac{\partial}{\partial x^1}+\frac{d}{dt}x^2(\gamma(t))\frac{\partial}{\partial x^2}$

Does this make sense or have I lost something? Also, in the statement of theorem 17.4, is it really needed to specify that the manifold is oriented, as we are restricting to a framed open set of it? thanks!