I live in a band E house. It's by no means a big house, being a modest 3 bed. It's not appropriate to assume all band E-G houses are mansions. We already pay an extortionate amount of council tax, with these bands already being subjected to larger rises than others a fee years ago.
It's not as simple as suggesting people move too- mortgage rates now may we'll mean paying more pcm for a smaller house, just to save on council tax.