It's easy to select capacitor models which fairly closely match reality.
The good thing here is that simulations will tend to use lower ESR capacitors, than what's available, therefore model the worst case.
It appears the class-B output stage was what was causing the instability, as the op-amp had a step change in its output, when changing from sourcing to sinking and back again. The above circuits are more class AB, which is more stable, but it's probably still prudent to test with different capacitor and ESR values, at different currents.