The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere are open.
Is this a triviality? The geometry of the embeddings can be non-trivial, so I doubt that a CW-structure/deformation retraction is so obvious (for Hatcher, Rotman etc.) that they don't even mention it. Note that the image of an embedding is not necessarily closed/open.
Even worse, this fact is used by Hatcher to prove invariance of domain itself! Can someone illuminate for me this problem leading to a (seemingly) circular reasoning?
Hatcher has a proof that $\tilde H_i(S^n-h[D^k])=0$ for all embeddings $h:D^k\hookrightarrow S^n$ of balls into spheres and all $i\ge 0$. It can be found in chapter $2.B$ of his book Algebraic Topology. Note that $h$ must be a closed map since $D^k$ is compact and $S^n$ is Hausdorff. Hence $h[D^k]$ is closed and we can apply Mayer-Vietoris to the cover $A=S^n-h[D_-^k]$, $B=S^n-h[D_+^k]$