One other issue is eating and drinking. A person with dementia may tell you they are eating and drinking (and they may believe it) but they may not be. My mother lost a lot of weight despite saying she was absolutely fine, and it was at that point I got carers in to ensure she had two meals a day. They prepared the meals - but she also needed prompting to eat and drink, and needed to be supervised to ensure she did.
She also had other issues by that point - she couldn't remember how to use the washing machine, and she occasionally got lost at the local shops. And she regularly lost/forgot her keys and shut herself out. She wasn't actually setting fire to the flat or having regular falls, but she couldn't complete basic every-day tasks and was repeatedly asking help from neighbours.
Edit - sorry I was answering a slightly different question. The above applies to when she could not be safely left to live independently, and started having carers coming in for several hours. When she needed supervision 24/7 the issue which tipped the balance was wandering - going outside and becoming lost, and increasing reliance on neighbours to bail her out.