I think it is quite normal for people with dementia to have very little/no understanding about the reality of their situation.
Due to her poor mobility and not being able to go up and down stairs it would not be safe for her to be at home where she would probably try to go up/down stairs and then fall. Im sure that deep down you know this, but it is hard when a parent is begging you to take them home.
My mum also went from hospital to a period of rehab/assessment in a care home as part of a reablement scheme. I told mum she was convalescing in a much nicer place than hospital and she would be going home once she was better - this is known on here as "love lies". Mum seemed to accept this. She did not actually improve, though, so at the end of her reablement she stayed there permanently, but by this stage she had settled.
Do check up about her complaints, but I must say that in my experience, people with dementia will make all sorts of claims - the staff mistreat them, everyone ignores them, they dont get fed, etc, etc, when none of it is true. Mum showed me a bruise on her arm about the size of a thumb print which she insisted was caused by a carer pinching her "very hard". In reality I had been sat next to her the day before when the nurse did a blood test and this was the resulting small bruise!