Dio Sabbath is *basically* a continuation of Rainbow
I do not believe it's an incredibly outlandish thing to say.
Dio brought magic and dragons and knights and all of his stock harmonies to Black Sabbath.
***I am not saying those are bad or undesireable things***
However, there can be little argument that he brought these things directly from Rainbow as his tricks of the trade.
The remaining Sabbs adjusted to rise up to what Dio was accustomed to.
If you pulled someone off the street and played them LLRR and then played them HH and asked them if they were different bands, a good deal of them would say no.
Blackmore is a bit more technical and flashier, Iommi is obviously heavier.....but overall, it's the same vibe with the same voice and the same worldview.