I suppose it depends a little on how the current carer feels about employing someone to take on the nights. Would this be enough to make them less tired? Would they (the current exhausted carer) be going back to their own house to rest and sleep, or would they be staying put and if so is the house layout such as to allow all the night time caring to be taking place without disturbing the sleeping person?
My own thought about care at home v residential care was that residential care offers many hands and many eyes, so no one carer is responsible for hours at a time and there is less opportunity for... less than good care, if you know what I mean. I didn't like the idea of my mum being entirely dependent on one person and behind closed doors.
Also, at home care, especially night time care, is expensive.
Lastly, to be perfectly honest, no one wants to go off to a care home but given how poorly your mum is it's highly unlikely that she'll be aware enough to recognise a care home for what it is. My mum was still quite with it when she went into care but had no idea where she was. It became all kinds of places to her and at one point (and this I hear this is common) she thought she owned it. So I wouldn't worry too much about what your mum might think. Much better, as you say, to look at what she needs right now and how that might best be delivered, without making anyone exhausted.
I never thought I'd be saying it but my mum thrived in the care home and had a much better quality of life than when she was in her own home. She's still with us, no quality of life to speak of because of the advanced Alzheimer's, but the care she receives is still fantastic and it's probably the reason she is still with us. So don't be too afraid to go the care home route if you feel it's the right thing to do, for everyone.