Not true in most U.S. real-estate markets (especially in cities), where rental prices bear very little relationship to costs to the landlord. Prices are set more by a mixture of demand and supply, where supply is more constrained by zoning and planning policy than by maintenance costs. Raising or lowering property taxes in that kind of market will raise or lower profit margins for the landlord, but not raise or lower the prevailing rental prices, which are already set as high as demand allows.

One way to verify that not even landlords believe this "pass on the costs" theory is to see what they lobby for. U.K. landlords are adamant in lobbying against any attempt to convert the council-tax system into a property-tax system, and California landlords strongly lobby against any modification to Proposition 13. If it were equivalent either way with all costs just paid by the renters, landlords wouldn't care enough to lobby.

If rental income is not sufficient to cover property tax and other property maintenance costs, the property either gets sold instead of leased, or the owner is speculating on an increase in the value of the property and just rents to slow the bleeding until the market is willing to pay the target sale price. Self-storage businesses are often used for the latter purpose. If you see a self-storage out in the middle of nowhere, it usually means that someone is betting that area will see a development boom soon.

Property tax hits all landlords in an area in roughly equal measure, so it establishes a rental price floor without any of the illegal and impractical forms of collusion. Rental prices below that floor generally do not exist unless some kind of funny business is going on, because few people are willing to operate a business that continuously loses money without some kind of plan for getting it into positive cash flow.

Later on, if property taxes increase by a marginal amount, rental prices may increase by less than that marginal amount, but they will also eventually cause the cheapest rents to exit the market, as the tax increase causes them to cross the boundary of profitability, so further into the future will increase the surviving rents due to contraction in supply, and possibly increase in demand, depending on what the property taxes get spent on.

As such, if you are a land-owner in the US, and own property that rents at above-median rates, it would benefit you to lobby the local government to increase property tax rates, improve the public school system, and invest in public infrastructure. This tends to force out your cheaper competition, who might then sell out to you or convert to condominiums, and attract new tenants willing to pay higher rents. Tax increases come out of your tenants' pockets. But if you are at the cheapest end of the rent scale, you would likely prefer to lobby for zoning and occupancy restrictions, and to keep property taxes low. Your tenants have a hard limit on what they can afford to pay, and so the tax increases come straight out of your pockets. So there are valid reasons for landlords to lobby, both for and against.

Council tax is paid directly by the tenant, who is liable for it personally, and whose name appears on the demand.

Not exactly, most rent contracts require you to register with the council and pay the tax bill but there is nothing that prevents the property owner from paying the bills for you.

If you rent via an all inclusive scheme then you are not liable for it personally.

As long as someone is paying the tax and no discounts (single occupancy, benefits etc.) are applied on the tax the council doesn't really care who and how many people live on the property.

Renters can claim property tax refunds, at least in my state. I don't quite get why, to be honest, but the concept does exist.