I think that owners should be completely liable for council tax. It's a nonsense that they pass this on to tenants.
Why is it nonsense, any more than it is that tenants have to pay for their own electricity, gas, water, contents insurance, TV licence, broadband etc.?
You could go even further and say that, as the landlord owns the kitchen cupboards and any built-in bedroom furniture, they should also be expected to fill them with food and clothes for you. After all, why should you be expected to buy food to put in somebody else's cupboards - just because you are the one who will be eating it all?
The landlord will pay for the buildings insurance, because that's protecting their asset. The landlord will pay for repairs to the fixtures and fittings of the building, because this is maintaining their asset and also continuing to provide in good working order that for which you are paying the rent as agreed. The landlord will pay for any mortgage on the property, because that's helping them to buy their asset. They won't pay your council tax for you, which pays for all of the local services that you as resident will/may be using, as that has nothing to do with the property as an asset and everything to do with the property as a home.
Would you be happy for the landlord to have a key to your home and pop in whenever they like? Of course, not, because it's your home and not theirs.
Some rental properties may have one or more of these included in the rental cost, but your rent will be higher to take account of this.
If you do want all of your living costs (other than food and clothing) taken care of for you and to just have to pay one single bill in return for it, you're perfectly at liberty to book permanently into a hotel. Just don't expect it to cost the same for a month there as it would cost to rent a house or flat.