I do believe, from personal experience, and from what I read on TP, that eventually the first stage of living with dementia progresses to the next stage. In the first stage carers support independent living for as long as that person can manage this. Note that I say as long as the person with dementia can manage, not the carer. This is because the carer cannot halt the disease process, and neither can the person they care for. No matter how much love, effort and willpower is applied, damage to the brain keeps on going.
The next stage in caring is to ensure safety. The value of independence to them is now in the past. Fear and confusion increase and the person we care for makes regular cries for help. These take the form of: frequent phone calls, demands that you deal with the ever increasing domestic crises, sundowning and nightmares, knocking on neighbours' doors, wandering, complaints to the bank or police about imagined crimes, aggression towards caregivers, etcetera.
This person no longer feels safe living at home. What once was a haven is no longer so, and they set off physically or emotionally in search of their idealised 'Home'. Some people are able to manage at this stage with a suitable team of carers, both family and paid, to support them at home. The person with dementia now needs 24/7 supervision. This is where most home care plans fall down because paid night care is too expensive for the state to pay for.
This is when the person with dementia usually needs to relocate to a residential care facility. The benefits here are better resources for personal care and social activities, laundry and meals provided, dementia trained staff (not always!), teams of staff whose energy is not drained by unreasonably long caring hours. The new resident may demand rescue from being looked after, but it's really the dementia that is imprisoning them. The CH is a place of safety now, which their former home out in the world can no longer be.
I am sure I'm not telling you much that you don't already know. I just wanted to lay it out before you to indicate that, from what you write, the first stage of dementia care for your dad is already over. He needs effective team care 24/7. My mum has received this in her own home for over 6 years. My MIL has received this in a CH for nearly a year. The difference between them is mainly financial. Well, that's not true. They have very different personalities, life experiences, and different dementia behaviours and challenges. What I meant was that for my mum home care was affordable, while for MIL it was not, once the family team's efforts and availability no longer met her needs.
However, MIL would never have accepted live-in carers. She is physically active and would have run away, locked them out and probably yelled and hit out at them in her frustration. She attempts to do all that now, but obviously the CH staff have greater control than they would in a normal domestic environment. My mum has been horrible to some of her carers with bullying, feigning illness, refusal to get out of bed, etcetera. She's now a pussycat and enjoys her care, but while she still had fantasies of getting her independence back she could be pretty manipulative.
If you and your dad are not yet ready for him to move to residential care you could consider employing a live-in carer for a couple of weeks, just to see if this might work longer-term. It is not cheap. Currently my mother pays £600 p.w. for carer's charges and travel, plus another £100 p.w. in care agency fees. This is per carer, she now has 2 at once, but managed with one at a time for several years until her night care needs made a significant impact on the ability of one person to manage her needs 24/7 while getting adequate sleep.