It's also not even clear that Saxon is actually the origin for English; yes the Anglo-Saxons often called themselves Saxons, but it is likely the ethno-linguistic pool they came from was something closer to Frisian than Saxon. Or some sort of Common West Germanic, from what is now Schleswig-Holstein and Frisia, not Saxony proper.

So yeah there would have been mutual intellegibility, because at that point there was likely high mutual intelligibility between all Western Germanic languages. But like you said that doesn't make them the same language. And like you aluded to, it's kind of a lumpers vs splitters thing.