OP, "Go Home" is said by the elderly as ... I wish to be as I use to be. Read up on it. He's very unlikely to walk out the door and try to go to some place you're thinking of as his former home. No, if he's not medically capable, no reasonable facility will knowingly let him walk out the door. Of course not. If he leaves, runs away, he likely would be found and I bet after once he wouldn't try that again.

PP. but they won’t actually discharge him without a safe place to go.

Can leave whenever they choose unless they have been declared incompetent. If they don’t return when expected, likely the nursing home will issue a silver alert. And often staff will try to talk residents out of leaving. But in the end, they are residents and not prisoners.

Has his family taken steps to become his legal guardian and/or have medical power of attorney? I don't know what the exact terms are, but when my dad with dementia went into the care facility he regularly asked to leave but he was not allowed to because it was my mother's choice at that point.

They should seek advice on this if they have not.

Unless he’s there by court order he can leave whenever he wants.

They can if he goes AMA (against medical advice).

This elderly person cannot live independently (can’t shower on his own, is a fall risk, incontinent) but wants to “go home.” Only he has no home to go to. He doesn’t own a place and can’t afford to rent a place. Can the nursing home just discharge him with no place to go?