The Covariant Derivative as a Derivation on $\mathrm{TFA}^{\infty}({M})$

Question

Let ${M}$ be a smooth manifold, then let $\mathrm{TFA}^{\infty}({M})$ be the smooth tensor field algebra, which is defined as \begin{align*} \mathrm{TFA}^{\infty}({M})=\bigoplus_{{m}={0}}^{\infty}\bigoplus_{{n}={0}}^{\infty}\Gamma^{\infty}({M},{T}{M}^{\otimes{m}}\otimes{T}^{\star}{M}^{\otimes{n}}){\,}{.} \end{align*} Can one then simply define the covariant derivative $\nabla$ as a derivation $\nabla\in\mathrm{Der}(\mathrm{TFA}^{\infty}({M}))$, more precisely as a mapping \begin{align*} \nabla:\Gamma^{\infty}({M},{E}{M})\longrightarrow\Gamma^{\infty}({M},{E}{M}\otimes{T}^{\star}{M}){\,}{,} \end{align*} where ${E}{M}$ is some tensor bundle over ${M}$, which behaves as an extension of the exterior derivative of smooth scalar fields $\mathrm{d}:{C}^{\infty}({M})\longrightarrow\Gamma^{\infty}({M},{T}^{\star}{M})$ without the necessity of considering it as a directional derivative?