I walk through The East Cut nearly every day, and I admit I was confused by the rebranding. But it wasn't done by Google. There are people cleaning the sidewalks in East Cut shirts, banners that say East Cut, real estate companies that call it East Cut. It would be a little weird if Google ignored the name.
Yes, "The East Cut" is a strange name. But that area as never part of South Beach (which is also a strange name given that there is no beach). It wasn't really Rincon Hill, it is the area below Rincon Hill and is at sea level.
The East Cut is part of the larger South of Market area, and some of it could be considered to be the southern part of the Financial District[1]. But the neighborhood has evolved into its own thing, and deserves its own name.
[1] https://en.wikipedia.org/wiki/List_of_neighborhoods_in_San_F...