This is not true, the local authority have to accept your application for council tax support, but it's not paid by them, it is paid direct to tenants as part of the universal credit housing payment.
@Gingerrogered, no, I am sorry you have got it wrong. Under UC, you apply for a council tax reduction. If entitled, you are not billed for the amount you are entitled help with. It’s the same principle as how they do it for JSA and ESA. However, this time you will have to apply for the council tax reduction whilst with JSA and ESA it’s worked out when you apply for housing benefit.
www.gov.uk/apply-council-tax-reduction