In Lee's Introduction to Smooth manifolds (earlier version), he proves the Darboux theorem using
Lemma. Let $(M, \omega)$ be a symplectic manifold, and let $(U;x_1, \ldots , x_n, y_1, \ldots, y_n)$ an open chart. Then $(x_1, \ldots , x_n, y_1, \ldots, y_n)$ are Darboux coordinates if and only if $$\{ x_i, x_j \} = \{ y_i, y_j \} =0 \qquad , \qquad \{ x_i, y_j \}= - \delta_{ij} $$ Lemma. Let $M$ be a manifold and $X_1 , \ldots , X_r$ linearly independent vector fields. Then there exist coordinates $(u_1, \ldots, u_n)$ such that $X_i = \frac{\partial}{\partial u_i}$ if and only if $[X_i, X_j]=0$.
He uses induction on $k=0, \ldots , n$, and claim "for $k=0$ there is nothing to say", and start proving the general case. But I don't feel really comfortable with that, since $k=0$ is literally nothing.
So I would like to see at least the case $k=1$. But my problem arises when I discover that the arguments he uses for the general case are not valid for $k=1$. So, how to do it? How to obtain functions $(x_1, y_1)$ such that $\{ d_p x_1, d_p y_1 \}$ are linearly independent and $\{ x_1, y_1 \}= -1 $?
This isn't a proof of your lemmas, but should answer the question in your last paragraph.
Claim. Let $(M,\omega)$ be a symplectic manifold and take $p\in M$. There are functions $x,y$ defined on a neighborhood $U$ of $p$ with $\{x,y\}=-1$.
(Proof.) Take a coordinate neighborhood $(U;u_1,\ldots,u_{2n})$ of $p$ and set $y=u_1$. Now $\omega$, being nondegenerate, induces an isomorphism $T^*M\to TM$ which produces for each 1-form $\alpha$ a vector field $X_\alpha$ so that $\iota_{X_\alpha}\omega=\alpha$. Applying this isomorphism to $dy$ gives us a vector field $X_y$ for which \begin{equation} \iota_{X_y}\omega=dy. \end{equation} Now let $x\colon U\to\mathbb{R}$ be such that $\partial_x=-X_y$. Then \begin{equation} \{x,y\} = \omega(X_x,X_y) = dx(X_y) = dx\left(-\partial_x\right) = -1, \end{equation} as desired.
I'd also recommend reading a proof of Darboux's theorem using the Moser technique.