It's the NHS rather than social services that needs to accept the costs unfortunately nomore that's what the problem is.
Social Services have systems and processes in place for this kind of situation because it's a very common situation for them to manage. Social services employ social workers to manage and broker care packages in people's homes, check care plans are working, provide support to carers etc. They have systems in place to to approve one off, urgent or bespoke packages of care like this.
The NHS often doesn't have the same systems in place because they work on a much more structured and hierarchial/institutionalised model so other than district nursing type provision they don't have a huge amount of 'community care' services at their immediate control and disposal, and due to it being the NHS it's often a slow process to set up and broker new contracts with care agencies. The people in charge of 'case management' (all the 'stuff' that makes lives easier at difficult times) are nurses whose background and expertise is clinical rather than social welfare, and their role is usually to assess, award funding and then move on to the next person, not to continue to manage packages of care in people's homes. They simply don't have the resources, skills or time.
The NHS manages this as best they can usually by having contracts with a few agencies yand give all of their 'domiciliary CHC' work to those few agencies, but that system breaks down if those agencies don't have the capacity to cover urgent care packages or if they are required quickly in areas they don't already have staff.
On top of that there are huge care staff shortages in many areas anyway.
Sadly the NHS only have a duty to provide and fund care, so if they can't find an agency they will probably try and push the family towards residential care instead. There's pretty much zero chance they will pay a non CQC registered live in carer.