In most cases it did.But it didn't work that way !
Working class people living alone generally ended up paying less than they did in rates, most working class (and many middle class I expect) families of two adults ended up paying more, more than two adults in the house and it was considerably more.
However, once your house was beyond a certain size, even with many adults in the house you were better off under the poll tax system.
There will always be the argument that a large house doesn't necessarily mean you are well off, but generally property size/value is reasonably proportional to income.
Yes, some will always argue that everyone should pay the same for public services, but a large community bill hits people on lower incomes hard, while people on high income barely notice it.
The idea of modern society is that people are looked after, although as said, that wasn't the idea of Thatcherism...no such thing as society...