Shifting health care funding from central government to local councils is unfair to low paid workers. People who fall below the income tax threshold but still pay council tax. Increasing council tax to pay for local health care puts an extra burden on low income earners while relieving that burden from high paid people.
The only fair taxation is based on how much you earn.
Council tax is not. Income tax is.
Nobody seems to be raising this point when they discuss health care funding on the news.