It's a political choice: whether to have a taxes based on house size, or whether to have a poll tax - or both. How high to set income tax, whether to have a wealth tax etc.
My point is that having a tax based on house value (whether council, mansion, or a wealth tax which is only on housing rather than total wealth) is unfair in a country with so much house price disparity. It's also overly complex and expensive to administer.
Our current council tax system was also based on house prices (now out of date) and that's also wrong. Admittedly there wasn't the same house price disparity when it was first introduced, so it wasn't quite as unfair as it will be if it's recalculated on current house prices.
Service usage is a reasonable thing to charge for - in which case a poll tax makes sense. And adjust income tax, if you want to continue to soak the rich more - at least it's honest.
Paying for social care as part of council tax was probably an attempt to make councils more responsible - by making them face the music for their spending. But councils are pretty incompetent at money management, so that backfired.
I also actually think social care is a cost which should be shared across the country. But honestly, as a clear tranche of government redistribution.