In famous The Index of Elliptic Operators: I Atiyah and Singer introduce two families of morphisms: $$\text{a-ind}^X,\, \text{t-ind}^X\colon K(TX)\to \mathbb Z$$ indexed by compact smooth manifolds $X$. The index theorem says that $\text{a-ind}^X = \text{t-ind}^X$.
However in the proof they define an excision property that essentially allows one to define analytical and topological indices for every (non-compact) smooth manifold, basically by using any embedding into a compact manifold.
My understanding is that we can write $\text{a-ind}^X = \text{t-ind}^X$ for every (not necessarily compact) manifold $X$.
Have I missed an important point? Is there any reason why the theorem wasn't formulated as this?
You missed an important point. Excision holds under open embeddings into compact manifolds – and this is not always possible.