I don't see why it is cheaper. In a care home they would have one carer on minimum wage looking after a number of patients (say 6), so how can that be more expensive than one-on-one care. I don't understand the logistics. I suspect you are right but it doesn't make sense.
Is your live-in care 24 hours? I think if the carer has to get up during the night then the costs increase. Also what about moving the caree (I presume they are quite mobile and can even get dressed), will it require two carers at some point or do they rely on equipment (hoists, wheelchairs etc)?
I expect it is good to have the loved one stay at home but space is needed for the carer and they would need certain facilities (wet room...perhaps?). So perhaps it isn't for everyone.