I think the same proof works in general (for closed manifolds). An operator is Fredholm when it is invertible up to compact operators, and an elliptic operator admits a parametrix. So when we're working over compact manifolds, the parametrix gives us invertibility up to compact operators. Here we use the Rellich–Kondrachov embedding theorem to get compactness for $W^{k+m,p}\to W^{k,p}$ (the literature typically focuses on $L^2$ for using "Rellich compactness" and Hilbert space machinery).

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). We need asymptotic conditions (or boundary conditions, there is a way to pass between the two) otherwise the kernel of our operator is infinite-dimensional. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's *"Monopoles and 3-manifolds"*, specifically Chapter 17.