I am confused by an exercise in Huybrechts Complex geometry: an introduction regarding sections of a certain holomorphic vector bundle over a Kähler manifold, related to Hodge theory. The exercise is
Exercise 4.1.4. Let ${E}\rightarrow X$ be a Hermitian ($h$) holomorphic vector bundle over a compact Kähler manifold $(X,\omega)$. Show that any holomorphic section $s\in H^0(X,\Omega_X^p\otimes {E})$ is harmonic.
The key definition is that of harmonic. As in the book, a (smooth!) section of the (smooth!) vector bundle $\Lambda^{p,q}X \otimes E$ is said to be harmonic if it is in the kernel of the Dolbeault-Laplacian operator $$ \Delta_{E} = \bar{\partial}_E\bar{\partial}_E^* + \bar{\partial}^*_E \bar{\partial}_E, $$ where $\bar{\partial}_E$ is the Dolbeault operator which encodes the holomorphic structure of the bundle $E$ and $^*$ refers to adjoint with respect to the $L^2$ inner product on smooth $E$-valued (p,q)-forms $$ (\alpha,\beta)_{L^2} = \int_X (\alpha,\beta)_{h,\omega} \text{vol} $$ which is inherited from the Hermitian structures on $E$ and $\Lambda^{p,q}X$ the smooth complex bundle of (p,q)-forms.
Note that $\Omega_X^p$ is, as in the book, the holomorphic vector bundle of holomorphic p-forms, i.e., those (p,0)-forms which are $\bar{\partial}$-closed.
My previous and seemingly naive attempt:
If $s\in H^0(X,\Omega_X^p\otimes E)$ is a holomorphic section, it is necessarily a section of the smooth vector bundle $\Lambda^{p,0}X \otimes E$. We can check the condition locally, and it is sufficient to check that $s$ is $\bar{\partial}_E$ and $\bar{\partial}_E^*$ closed. The latter is easy, as $s$ is an $E$-valued (p,0)-form, $\bar{\partial}_E^*$ kills it since there are no $(p,-1)$-forms. To check that $\bar{\partial}_E$ also kills $s$, we reason locally.
We aim to impose holomorphicity of $s$, we need to check that it is annihilated by the Dolbeault operator of the tensor product! Since the Dolbeault operator of the bundle of forms is simply the usual Dolbeault operator on the base $\bar{\partial}$, we have that $$ \bar{\partial}_{\Omega_X^p \otimes E} = \bar{\partial} \otimes 1 + 1 \otimes \bar{\partial}_E $$ and applying to a section of the form $s=\alpha\otimes \varphi$ for $\alpha$ a (local) p-form and $\varphi$ a (local) section of $E$ we obtain $$ 0 = \bar{\partial} \alpha \otimes \varphi + \alpha \otimes \bar{\partial}_E \varphi\qquad (*) $$ What I am missing is (1) how to use this to deduce that the extension to $E$-valued higher-forms of $\bar{\partial}_E$ annihilates $\alpha\otimes \varphi$. Recall that this extension gives a $p$-graded Leibniz rule $$ \bar{\partial}_E(\alpha\otimes \varphi) = \bar{\partial}\alpha \otimes \varphi +(-1)^p \alpha\otimes \bar{\partial}_E \varphi $$ I suspect I have a deep misunderstanding regarding this. (2) Where do I even need the Kähler condition that Huybrechts uses in the statement of the problem?