I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\infty$-categorical equivalences as weak equivalences, are equivalent to topological spaces, with weak equivalences being weak homotopy equivalences.

Now for simplicity, let me focus on the version of the homotopy hypothesis that says that $1$-groupoids, with categorial equivalences as weak equivalences, are equivalent to topological spaces, where weak equivalences are weak $1$-equivalences, ie. maps which induce isomorphisms on $\pi_0\,,\pi_1\,.$

Now on the other hand, Lie's second and third theorems imply that there is an equivalence of categories between simply connected Lie groups and Lie algebras. These theorems generalize to Lie groupoids and Lie algebroids, where simply connected becomes source simply connected (well, in order for Lie's third theorem to hold completely one needs to use smooth spaces which are generalizations of manifolds, but we can always replace "Lie algebroids" with "integrable Lie algebroids", in any case).

Right now these two results may not seem related, for two reasons:

There is no Lie algebroid present on the topological spaces. However, if we work with manifolds instead (or some appropriate class of infinite dimensional manifolds), then $M$ comes with a natural Lie algebroid, namely $TM\,.$ Therefore, there is a canonical Lie algebroid present.

The groupoids in the homotopy hypothesis are discrete, so there is no source simply connected condition. However, the source simply connected integration of $TM$ is the fundamental groupoid $\Pi_1(M)\,,$ and this is Morita equivalent to $\pi_1(M)\,.$ Therefore $\Pi_1(M)\,,$ with the topology associated with the smooth structure, is equivalent to $\Pi_1(M)$ with the discrete topology, and I believe this is the correct groupoid to compare $M$ with in the homotopy hypothesis for 1-types. So in a sense, the source simply connected condition is naturally present.

Now, if we assume that we that we have a notion of weak equivalence of Lie algebroids which implies that a morphism $TM\to TN$ is a weak equivalence if the induced map $M\to N$ is a weak $1$-equivalence, then we seem to get a connection between Lie's theorems and the homotopy hypothesis, ie. the homotopy hypothesis for (smooth) 1-types seems to be implied by a version of Lie's theorems (this wouldn't exactly be Lie's theorems since Lie's theorems use isomorphisms as weak equivalences, but we can use other weak equivalences instead).

I can go into more detail, but has this connection been written about elsewhere in the literature, or is there any reason to doubt this connection?