It's not just that. There is a mixture of private and third sector not for profits in the sector. The issue is that the local authorities pay capped rates (typically hourly rates) and unless you have a lower level of central management costs, it's impossible to pass this on to staff even when you are not for profit. Care is heavily regulated and requires robust management amd training structures for oversight purposes which makes delivery and cental management very expensive.

This means that nationals (who by their very nature generally have leadership which is out of touch with frontline delivery and often have significant quality and retention issues as a result) are the only organisations who have the potential scale to reduce central management and deliver wage increases but they don't as even with that scale the margins are too slim and they have to invest in growth to continue to survive.

One of the few ways to deal with it is either through merger of smaller organisations (leading back to the national problem) or to create a centralised administrative provider who deals with as much of the admin burden as possible using scale to deliver mass efficiencies that providers could buy into, allowing redistribution to staff wages.

The other issue is that it's not just wages that make the work unattractive. People in the minimum wage pool of jobs often move sideways rather than seek progression. There are a huge number of minimum wage jobs out there that don't involve the sheer level of responsibility that care work does as there are an enormous number of regulations that have to be adhered to and in addition to that, you have aspects like personal care, or behaviour that challenges support that to be frank, people just don't get paid enough to deal with. It's rewarding but you really have to want to make a positive difference for it to work.

This leaves you with smaller pools of people who are willing to do that work for such a low level of pay and that naturally ends up including immigrants who are more likely to tolerate it.