I don't understand why we cannot define the pushforward using a local $C^1$-extension.
To be precise, let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $x\in M$ and $f:M\to\mathbb R$ be $C^1$-differentiable at $x$. Then, by definition, $$T_x(f)v=(f\circ\gamma)'(0),\tag1$$ where $\gamma$ is an arbitrary $C^1$-curve on $M$ through $x$ with $\gamma'(0)=v\in T_x\:M$.
That's fine and I know how we can show that $(1)$ does not depend on the choice of $\gamma$. However, let's take a step back: By assumption, $$\left.f\right|_{O\:\cap\:M}=\left.\tilde f\right|_{O\:\cap\:M}\tag2$$ for some $\mathbb R^d$-open neighborhood $O$ of $x$ and some $f\in C^1(O)$. Now let $(I,\gamma)$ be a $C^1$-curve on $M$ through $x$ and $v:=\gamma'(0)$. Since $\gamma$ is continuous and $\Omega:=O\cap\partial M$ is $\partial M$-open, $\gamma^{-1}(\Omega)$ is $I$-open and hence $$\tilde I:=(-\varepsilon,\varepsilon)\cap I\subseteq\gamma^{-1}(\Omega)\tag3$$ for some $\varepsilon>0$. Most importantly, $\tilde\gamma:=\left.\gamma\right|_{\tilde I}$ is again a $C^1$-curve on $\Omega$ (hence on $M\supseteq\Omega$) through $x$ and hence $$T_x(f)v=(f\circ\gamma)'(0)=(\tilde f\circ\tilde\gamma)'(0)={\rm D}\tilde f(\tilde\gamma(0))\tilde\gamma'(0)={\rm D}\tilde f(x)v,\tag4$$ where ${\rm D}\tilde f(x)$ is the ordinary Fréchet derivative of $\tilde f$ at $x$.
So, ${\rm D}\tilde f(x)v$ does not depend on the choice of $\tilde f$ (since the left-hand side of $(4)$ does not). What am I missing? Why don't we define $$T_x(f)v:={\rm D}\tilde f(x)v\tag5$$ for any $C^1$-extension $\tilde f$ of $f$ at $x$ and $v\in T_x\:M$?
Definition 1: Let $E_i$ be a $\mathbb R$-Banach space and $\Omega_1\subseteq E_1$. Then $f:\Omega_1\to E_2$ is called $C^1$-differentiable at $x_1$ if $$\left.f\right|_{O_1\:\cap\:\Omega_1}=\left.\tilde f\right|_{O_1\:\cap\:\Omega_1}\tag0$$ fomr some $\tilde f\in C^1(O_1,E_2)$ for some $E_1$-open neighborhood $O_1$ of $x_1$. $(O_1,\tilde f)$ is called $C^1$-extension of $f$ at $x_1$.
Definition 2: Let $M$ be a subset of a $\mathbb R$-Banach space and $x\in M$. $(I,\gamma)$ is called $C^1$-curve on $M$ through $x$ if $I\subseteq\mathbb R$ is a nontrivial interval with $0\in I$ and $\gamma:I\to M$ is $C^1$-differentiable with $\gamma(0)=x$. Let $$T_x\:M:=\{\gamma'(0):\gamma\text{ is a }C^1\text{-curve on }M\text{ through }x\}.$$