Is there a simple proof/counter-example for this?
If we have a (big) connected graph $G$ with a big stable set $S$ and with bounded maximum degree (read "small"), then there's a long induced path (in $G$) containing a large number of the vertices from $S$ or there's a large subset of $S$ which contains only leaves of $G$.
While it seems natural, I've been staring at it for an hour and now I'm starting to believe it's false. Thanks!
Let $G_0$ be a triangle. At the next step create $G_1$ by attaching a triangle to each vertex of degree 2 (so $G_1$ has 9 vertices, 3 of degree 4, 6 of degree 2). At each next step attach a triangle to each vertex of degree 2, so the maximum degree will never exceed 4.
Taking one vertex of degree 2 from each triangle you added in the last step you can see that $G_n$ has an independent set $S$ of size $3\cdot2^{n-1}$ (it has larger independent sets, but this one will do). The graph $G_n$ has no leaves at all and you can easily see that any path can contain at most 2 of the vertices of $S$. So the statement, indeed, is not true.