There are quite a lot of assumptions on here and a fair bit of misinformation.
In England at least, very few Councils own and run their own care homes these days. They agree contracts with local, privately owned homes at discounted rates. The homes provide the same level of care for council funded residents as self funders, with self funder fees set at a higher rate so that the care home owners can off-set the discounts to Council funded residents. Many of these homes offer excellent care and are assessed by CQC with the same level of expectation as any other home.
If you are council funded, you will have gone through an assessment to understand your care needs before admission to a home (unless it is an emergency admission, at which point your assessment will be done in the home). Wherever possible the Council will aim to support care at home, even if self funding, as this is often the best thing for the person. Plus it is illegal to remove someone to a care home without their consent. The financial assessment is entirely separate and takes place after the care assessment.
For those people without sufficient funds to fund their own care, but levels of need which mean they can't remain at home safely, they will be offered a care home on the Council contract list. If (and only if) family can afford to and want to pay to top up fees so that a loved one can stay in a non-Councul funded place, they will be in a position to choose from a wider range of care homes. By this point you would be paying extra essentially for hotel costs and potentially higher staffing ratios as the care home will need to demonstrate it can meet the level of care needs for the person in question before you move them in - otherwise you may find they ask you to find another home more suited.
Most care homes - even the expensive ones - run on a mixture of core staff and agency staff, which are almost always paid peanuts for the work they do. And some of the more expensive homes are truly awful once you look past the hotel aspects of the care, often having much lower staff ratios and refusing to deal with increasingly complex needs of residents as they age. I had a great Aunt that spent a fortune on an extra care apartment with the intention of going into the nursing wing, when the time came. 4 years in, the very affluent owners decided to close the nursing wing and all that was left were the extra care apartments but without any care. In the end the Aunt had no money left and had to move into a Council home which could deal with her increasing needs due to dementia.
Nursing care is a different category altogether and has other types of funding routes associated with it.