Let $M,N$ be smooth manifolds, of which $M$ is compact. It is a well known result that for any embedding $f:M\to N$ there is some neighborhood $\mathcal{W}$ of $f$ in the Whitney topology such that any $f'\in\mathcal{W}$ is isotopic to $f$ and that the isotopy in question is covered by an diffeotopy of the ambient space (Th 2.4.2 in C.T. Wall's book), which can be assumed to have compact support, so that in particular there exists a compactly supported diffeomorphism $g$ of $N$ such that $f'=g\circ f$. The combination of these two steps is presented in Prop 4.4.4 of Wall's book. Now, by examining the proof of the results above and thinking of the velocity of the time-dependent vector fields involved I have convinced myself (hopefully not making a gaffe here) that for any neighbourhood $\mathcal{V}_{0}$ of $Id_{M}$ in the uniform convergence topology there exists some neighborhood $\mathcal{W}_{0}$ of $f$ in the uniform convergence topology such that if additionally $f\in\mathcal{W}\cap\mathcal{W}_{0}$, then the diffeomorphism $g$ above can be taken in $\mathcal{V}_{0}$. This is very weak, though.
My question is whether by restricting $\mathcal{W}$ in the Whitney topology one can ensure $g$ (in fact, the covering diffeotopy at all times) can be taken in any given neighborhood of the identity in the Whitney topology.
Related to this is the question of whether the group of compactly supported diffeomorphisms of a smooth manifold with the Whitney topology satisfies some form of semilocal path-connectedness. Namely, that for any neighbourhood $\mathcal{V}$ of the identity in the Whitney topology there exists another one $\mathcal{V}'\subseteq \mathcal{V}$ such that any element $g\in\mathcal{V}'$ can be joined to the identity by a path in $\mathcal{V}$.
I am quite confused, because after learning from Peter Michor's reply here that $Diff(M)$ is not locally path connected in case $M$ is open, I found Proposition 1.2.1 in Banyaga's book on diffeomorphism groups, which (vaguely quoting Omori or possibly Sergeraert) claims that both $Diff^{r}(M)$ and $Diff^{r}_{c}(M)$ are locally contractible, after very explicitly asserting a few lines above that $r\leq \infty$. I but be misreading something, but I cannot figure out what.