I'm currently reading John Baez's Gauge Fields, Knots and Gravity for fun, and I stumbled across the following when talking about connections in a vector bundle. So, let $\pi:E\to M$ be a smooth vector bundle, and $\nabla$ a connection on $E$. If we take a vector field $X$ on $M$ and a section $s$ of $E$, and express them locally over some open $U\subset M$ as $X=\sum X^i\frac{\partial}{\partial x^i}$ and $s=\sum s^je_j$ (for some local sections $e_j$ of $E$), then we get \begin{align} \nabla_Xs &= \left(X^i\frac{\partial s^k}{\partial x^i} + \Gamma^k_{ij}X^is^j\right)e_k \end{align} (in the book he uses the notation $A_{\mu j}^i$ instead of $\Gamma$'s and calls them the vector potential for reasons I haven't fully understood yet, but that's besides the point). Then, he goes on to talk about how to interpret $\Gamma$ more geometrically. He says the term $\Gamma^k_{ij}X^is^je_k$ is a local section of $E$, and is $C^{\infty}(U)$-linear in $X^i$ and $s^j$, which means it should be interpreted as an $\text{End}(E)$-valued 1-form over $U$, that is, a section of \begin{align} \text{End}(E|_U)\otimes T^*U. \end{align} Then, later on he says this is not just a local description, but in fact global, so $\Gamma$ is a section of $\text{End}(E)\otimes T^*M$.
Here's where I'm getting confused. If we take $E=TM$ to be the tangent bundle, then it means $\Gamma$ is a section of $\text{End}(TM)\otimes T^*M\cong TM\otimes T^*M\otimes T^*M$. But doesn't this say that $\Gamma$ is a $(1,2)$-tensor field over the base manifold $M$, which goes against everything that is emphasized in an introductory Riemannian geometry course?
I must obviously be misinterpreting what is written in the text, or have made a mistake unwinding the various "abstract" definitions. Any help in clarifying this is appreciated.