They have a mimimum of 2 people, training, hoists and a electric hospital beds which can be raised up to a safe work hight, usually with bed guards which the paitent can use to assist. They also have specialist mattresses to prevent bed sores. Hospital rooms and hospital furniture are designed as a work environment, homes are not.
Lie down in the floor and try to roll on your side without using any of your legs. Then roll on your 'good' arm without using the 'bad' one.
You will have an ordinary low bed, no training, and he can't roll safely with the broken arm and he will have to be able to roll and hold to allow wipe time. OT's will be risking their job by giving you a quick training guide and you risk harming yourself and your dad.
Stop focusing on getting him home because if you take him home you run the very real risk of him never getting out of bed again.
How are you going to get him home if you cant even get him into a car?
You need the hospital physio to establish his base line physical capabilities.
The OT will "only" sort out some of the equipment after it has been recommended as needed by the physio. So they should be working as a team.
Can i suggest that you phone his GP explain that he is in hospital and that he has failed and see if they will issue a letter detailing his level of fitness and past medical history and current meds as MereDintofPandiculation points out it gives a baseline pre-hospital capacity with the added advantage this is a professional doctors opinion. Unfortunatly sometimes that is important.
I know my posts appear very negative but safely managing a full grown adult who is bed bound is a massive task. (You are at the start line of a marathon not at the finish line of a sprint.)
You are a serious risk of harm as is you mother.
He did not just stop walking. If you agree to a discharge they will not schedule extra tests. Keep saying "I do not understand why he is not up and walking" and ask what has been done to resolve that.