Given a fiber bundle $p:E\to B$ and a point $x\in B$, is the evaluation map $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ defined by $\varepsilon(\sigma):=\sigma(x)$ a weak homotopy equivalence when $\Gamma^0(E)$ is endowed with the compact-open topology? I'm having trouble coming up with a good counter example but also failed at assuming it was a weak homotopy equivalence in general and trying to prove it (which could simply hint at me being bad at this). Right now it seems unlikely to me that the evaluation maps are w.h.e.'s in general since surjectivity of the induced maps on homotopy doesn't seem to work in general since there are no global sections of $E$ in general, or am I completely off track right now?

EDIT: To make the context clearer I'm going to describe the particular scenario I'm struggling with right now. For a smooth fiber bundle $p:E\to D^m$ over the closed unit disc consider an open sub-bundle $p^k_{\omega}:E^k_{\omega}\to D^m$ of the $k$-jet bundle of germs of smooth local sections of $E$. What I'm trying to understand is why the restriction map $\rho:\Gamma^0(E^k_{\omega})\to\Gamma^0(E^k_{\omega}(0))$ is a weak homotopy equivalence when both spaces are endowed with the compact-open topology and $E^k_{\omega}(0)$ denotes the restriction of the bundle $E^k_{\omega}$ to $0\in D^m$. The reason why I formulated the question slightly differently above is because obviously $\Gamma^0(E^k_{\omega}(0))\cong (p^k_{\omega})^{-1}(0)$.

EDIT: After being convinced that the first paragraph has a negative answer in general I would like to restate the explicit question that I'm unable to answer: given everything from the second paragraph, why is the evaluation map $\varepsilon:\Gamma^0(E^k_{\omega})\to (p^k_{\omega})^{-1}(0)$ defined by $\varepsilon(\sigma):=\sigma(0)$ a weak homotopy equivalence?