You are completely right when you're in the position to think rationally. My only thought is that ... it's his Mom. I'm a married man with a few kids and am the sole income provider for my family (by choice not circumstance). I would move heaven and earth to ensure my family (my bride and children) were taken care of. Were I single, I'd live on the street if it meant the difference between my mom being tossed out of her home or me being able to prevent that.
It's completely irrational behavior from a Dave Ramsey standpoint (I'm a fan of his), and I know that elderly care isn't a choice between "stay in your home" and "suffer/die in an alley with dementia", but it's difficult to make sound decisions when dealing with such a serious emotional circumstances. (case in point, I don't even know the meaning of living on the street, but I know what decision I'd make regardless).
There's a bit of hindsight at play here. When I decided to move out, my dad did everything he could to convince me to buy a house rather than rent (throwing your money away! Tax deductions!). From a cash flow perspective, it didn't make sense. I had no savings and would have had to game the system via several different types of loans, but from the conventional wisdom of the time, it was: your house will never decline in value. I rented and I'm glad I did. I'm now a homeowner with a simple 15 year fixed that's nearly paid off.