The comments of @JHM and @Ivan Solonenko contain an answer to OP's question.

(1) The OP is correct that $j_0:=\phi_* \circ J \circ \phi^{-1}_*$ is an almost-complex structure on the image $\phi(U) \subset \mathbb{R}^{2n}$ which is compatible with the standard symplectic form $\omega_{std}$ satisfying $\omega=\phi^* \omega_{std}$, where $(U, \omega, J)$ is the initial pseudo-holomorphic structure.

(2) Since $\phi$ is a Darboux chart, then $\omega_{std}$ is also compatible with the standard almost complex structure $j_1$ on $\mathbb{R}^{2n}$ restricted to $\phi(U)$.

So we have two compatible a.c. structures $j_0, j_1$ on $\phi(U)$, and can even find a smooth $1$-parameter family of compatible a.c structure $j_t$, for $0\leq t \leq 1$.

(3) The OP constructs a complex vector bundle $E \to \phi(U) \times [0,1]$, where the fibre over $(p,t)$ is the $j_t$-complex vector space over $T_{p} \phi(U)$. Now the OP correctly applies a form of Hatcher's Prop.2, obtaining a complex vector bundle isomorphism $F: E|_{\phi(U) \times \{0\}} \to E|_{\phi(U) \times \{1\}}$.

(4) As clearly stated by @Ivan Solonenko, here the OP errs in interpreting $F$ as the differential of an isomorphism between the base space $\phi(U)$. Indeed $F$ is the identity map on the base $\phi(U)$. The existence of fibrewise complex isomorphisms does not imply that the base spaces $(\phi(U), j_0)$ and $(\phi(U), j_1)$ are holomorphic. If the isomorphism $F$ produced by Prop.2. was induced by a diffeomorphism between the bases $f: \phi(U) \to \phi(U)$, then we would have $F=df$ on the fibres, and would find $df\circ j_0=j_1\circ df$. But this is not the case.

In conclusion, fibrewise deformations like $F$ are not generally induced by maps on the base. I found this is a recurring point of frustration, especially in studying symplectic geometry, and is basically the subject of Gromov-Eliashberg $h$-principles. You might find Eliashberg--Mishachev's AMS textbook "Introduction to $h$-principle" interesting -- it contains several such examples as above.

The comments of @JHM and @Ivan Solonenko contain an answer to OP's question.

(1) The OP is correct that $j_0:=\phi_* \circ J \circ \phi^{-1}_*$ is an almost-complex structure on the image $V:=\phi(U) \subset \mathbb{R}^{2n}$ which is compatible with the standard symplectic form $\omega_{std}$ satisfying $\omega=\phi^* \omega_{std}$, where $(U, \omega, J)$ is the initial pseudo-holomorphic structure.

(2) Since $\phi$ is a Darboux chart, then $\omega_{std}$ is also compatible with the standard almost complex structure $j_1$ on $\mathbb{R}^{2n}$ restricted to $V$.

So we have two $\omega_{std}$-compatible a.c. structures $j_0, j_1$ on $V$, and can even find a smooth $1$-parameter family of compatible a.c structures $j_t$ on $V$, for $0\leq t \leq 1$.

(3) The OP constructs a complex vector bundle $E \to V \times [0,1]$, where the fibre over $(p,t)$ is the $j_t$-complex vector space over $T_{p} V$. Now the OP correctly applies a form of Hatcher's Prop.2, obtaining a complex vector bundle isomorphism $F: E|_{V \times \{0\}} \to E|_{V \times \{1\}}$.

(4) As clearly stated by @Ivan Solonenko, here the OP errs in interpreting $F$ as the differential of an isomorphism between the base space $V=\phi(U)$. Indeed $F$ is the identity map on the base $V$. The existence of fibrewise complex isomorphisms does not imply that the base spaces $(V, j_0)$ and $(V, j_1)$ are holomorphic. If the isomorphism $F$ produced by Prop.2. was induced by a diffeomorphism between the bases $f: V \to V$, then we would have $F=df$ on the fibres, and would find $df\circ j_0=j_1\circ df$. But this is not the case.

In conclusion, fibrewise deformations like $F$ are not induced by maps on the base. $F$ is not a differential $df$ except if integrability conditions are satisfied. For a.c. structures, the tensorial form of the integrability condition is given by Nijenhuis tensor. E.g., the $\pm i$-eigenspaces of a.c. structures $j|_p$ define distributions on $V$, and which are almost never integrable for generic a.c. structures.

This can be recurring point of frustration in studying symplectic geometry, and is basically the subject of Gromov-Eliashberg $h$-principles. You might find Eliashberg--Mishachev's AMS textbook "Introduction to $h$-principle" interesting -- it contains several such examples as above.