It looks like phase overrides are determined through a material which will have the properties you are looking for. So, if you collect all of the materials in the document, filter them to ones matching the specific names for phases, you can then get all of the material’s properties.
Something like this:
phase_names = ['Phase - Exist', 'Phase - Demo', 'Phase - New', 'Phase - Temporary']
materials = FilteredElementCollector(doc).OfClass(Material)
phase_mats = []
for mat in materials:
if mat.Name in phase_names:
phase_mats.append(mat)
Edit: This will not provide access to the Halftone property which is present for actual OverrideGraphicSettings. If the phase appearances are truly overridden then this will also not be reflected, but this strategy might get you closer to your goal.