I feel for you, but can only add that distracting her with a trip to a park nearby is something I did that my husband enjoyed for a an hour or so. Bring water and any appealing snack food.

'I want to go home' is so very common. I have come to believe it's actually them expressing that they feel confused and unsafe."Home" isn't necessarily a place. It's a feeling. I think that is what they're looking for.
My mum, once in a dementia unit, used to pack all her possessions every day. As the disease progressed, she used to go round the care home taking all the towels and putting them in her wardrobe. Then she started taking all the toothbrushes she cold find. Your wife's behaviour isn't unusual for someone with Alz, though to us it seems odd and probably irritating!

My husband did that at one period of time on his Alzheimer’s Journey. It was always late at night. One night I got a suitcase and he began to fill it. In the end he would usually get tired of collecting what he needed and would relax back in bed.