The writer is the mother of a girl with a debilitating and progressive disease, and she is speaking about how, as her daughter's illness degenerated, the hospitals supported her less; I understand that in that sentence she is saying that an all-care is claimed, but in reality the hospital doesn't give a caretaker, so the family has to step up for anything that's outside of the care given in reality.
But what's the meaning of 手前? I can't understand it as any of the meanings I found, so I'm not sure how it fits in the sentence; I also tried a monolingual dictionary, but still nothing comes to mind.
Not sure if this is part of the problem, I don't really understand the そうかといって part.
Edit: I just had a thought, not sure if it's right: since 手前 is also "in front of", could it be it means "in front of", but in a figurative manner? Like saying that on the surface they claim a complete care, but that's a facade and the reality is different.
Not sure about the grammatical structure, for example in that case I would expect a "but" of some kind ("On the surface they claim a full care, but"), but if that's the case it would make sense.