Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of Lipschitz vector fields on $M$ and $\exp:\mathfrak{X}(M)\rightarrow \mathrm{Homeo}_0(M)$ be the map taking a vector field to its integral curve (let's make similar notation for the $C^r$analogues when $r>0$).

The Chow-Rashevskii Theorem says that the flow associated to a vector field satisfying the Hormander condition can attain any point $y \in \mathbb{R}^d$ from any other point $x \in \mathbb{R}^d$. In other words, there exists some $X \in \mathfrak{X}(M)$ such that $\exp(X)(x)=y$.

*Does this imply density in the topology of pointwise convergence (point-open)?*Let $\{X_i\}_{i=1}^{\infty}$ be a collection of Lipschitz vector fields for which $\{X_j\}_{j=1}^i$ generate $d_i$ dimensional Lie sub-algebras $\mathfrak{g}_i$ of $\mathfrak{X}(M)$ (where $d_i<d_{i+1}$). This is a relaxation of the Hormander condition. Moreover, I ask that $\{X_i\}_{i=1}^{\infty}$ generates a dense linear subspace of $\mathfrak{X}(M)$. Moreover, $\exp|_{\mathfrak{g}_i}(\mathfrak{g}_i)$ defines a $d_i$-dimensional Lie subgroups of $Homeo_d(M)$ and therefore $ \cup_{i=1}^{\infty} \exp|_{\mathfrak{g}_i}(\mathfrak{g}_i) \subset Homeo_d(M), $ is "infinite dimensional". But when is its closure, in the compact-open topology, the entire space?

The result referenced in this answer states that the group generated by $\exp(\mathfrak{X}(M)) $ in $\mathrm{Diff}(M)$, we denote it by $\langle \exp(\mathfrak{X}(M))\rangle$, is $\mathrm{Diff}_0(M)$ the identity component therein. How can this be reconciled against this paper this result which shows that $\exp(\mathfrak{X}(M))$ is meager in the $C^1$ topology on $\mathrm{Diff}^1(M)$? I.e.: $$ \langle \exp(\mathfrak{X}(M))\rangle - \exp(\mathfrak{X}(M)), $$ is topologically non-trivial.

*Is there any topology stronger than pointwise convergence for which $\exp(\mathfrak{X}(M))$ is dense in $\langle \exp(\mathfrak{X}(M))\rangle$?*