My dad had middle stage dementia. This is past the "just a wee bit forgetful" stage but not quite at the "incontinent and bed bound" stage. Even though he was still at home he could not be left for 5 minutes. He confused day/night, didn't recognise people, wouldn't remember what he'd done 5 minutes before so couldn't be trusted to remember medication, needed prompting to do everything from going to the loo to putting his shoes on. Sort of like a toddler - except a big, strong toddler who can't just be put in their cot when they are having a tantrum. We wouldn't ever consider it safe to leave a toddler on their own all day with someone popping in morning and evening and it's simply not safe to leave people with dementia either. It's not a steady decline, it's up and down, good days and bad days, the evenings are way worse than the mornings.
https://www.dementiauk.org/information-and-support/about-dementia/stages-of-dementia/
It's really hard OP and I think you are probably underestimating just how much care he is going to need going forward. He is going to need someone with him pretty much 24/7. Yes you can do that with a combination of carers, family and other help but it is RELENTLESS and you do not get a break, ever.
I would not feel bad about lying to your Dad. It is really common for people with dementia to obsess over cost of things, and going home (although they mean their childhood home not their current home) on a loop because you reassure them, they forget, the worry kicks in again.
Your Dad is SAFE in a home and that is the most important thing, keeping him safe. Agree that getting social services or a medical professional to do an assessment is the way to go as in my direct experience family tend to downplay just how bad things have got.
When my Dad was ill I found the cockroach cafe threads on here a huge comfort, you are most definitely not alone in this situation.