Should he go to the hospital?

The fact that his back is in such bad shape shows that he is not getting proper care. If he was, it would be in better shape than that. Bed sores are usually just from what's happening to him laying in one position too long, not changing the bed sheets often enough and inattentive care. Proper bathing might not be happening as well even if it's just a sponge bath. He may not be cleaned properly or often enough. I had somewhat of the similar problem with my grandmother years ago. My bottom line on these nursing homes from my families experience is never trust them (completely) on what they say they are doing involving him.
Always visit at times they don't know when you're coming.
Don't have a set schedule or habit that they know when you're coming. Don't think they don't pay attention to this because they do.
If you cannot get the cooperation with them and they just keep sidestepping you with more excuses and you see nothing changing, I would see if you could get outside help to at first talk with (in total secrecy meaning the patient doesn't know and the establishment doesn't know) and then further action if necessary. I have seen inattentive abuse in these places whether they are being taken care of and paid for by Medicaid or the patients own finances.
Vigilance is required. I wish you all the luck because it can be difficult.

Idk what country you're in, but I work at a nursing home, and so many times we've sent patients to the hospital only to have them sent right back.. There's doctors and nurses at the nursing home, so unless he needs surgery or something they can't do there, they won't keep him for long.

I would ask for a meeting with the doctor and head of his department to find out what they're doing to help, you can also ask to see his journal (everyone that cares for your dad has to write in his journal anytime they check on him - so you can see there if they're following protocol). Ask the doctor to explain why they don't want to send him to the hospital, or ask for a second opinion.

Ofc you can also ask for him to be sent, but sometimes the hospital also refuses to take in a patient that's already in a caring facility, if they think he should be able to get the help he needs there.

Usually, they take care of everything in house at a nursing home, but if they're not you can insist he goes to the hospital.
They do send them out for emergency care.
It won't solve the problem, long term, if the nursing home is not taking care of him properly.
There should be documentation on those bed sores and I believe you can get those if you are his power of attorney.
Nursing homes have to take notes about patient condition and those kind of things.
If he does go out to the hospital and they do help whatever is going on once they send him back and he's not being properly cared for the problem will come back.
My husband was a CNA and this is the kind of thing that would drive him insane.
He always put his clients first.

Take him to the hospital yourself if possible. And I like Samuel's idea, call the ombudsman.

I've known people in and also worked in a few really horrible rehab centers. For some reason it's not uncommon to see this type of malpractice happen. The nurses are overworked but there are also those that truly just don't care about the patients and don't do their jobs correctly, they will sit around in their phones or whatever whenever possible. Hold them accountable.

Contact the institutions ombudsman. All such institutions are (were pre Trump) and are charged looking out for the welfare of the patients. It sounds to me that he is not receiving quality care!

The Nursing Home should have a Registered Nurse on staff at all times. Because of this the Hospital will not keep him for long. If they're not doing 4 hourly pressure turns on him. That is medical negligence. Especially now he has a pressure sore, it should be 2 hourly turns.
Apply pressure on management anyway you can to get your father the care he needs.

I wonder if one of those motorized beds that shift the patient when in a coma etc. will help him...

He might be eligible through Medicare if it's prescribed for him....

He needs to go to hospital ,for your own peace of mind as well. if anything happens you will blame yourself forever, "I know you," so please get him to a hospital for both your sakes.

Yes. He needs to go to the hospital because he is not receiving proper care. You have a good case for a lawsuit. The state views sores like that abuse.

Yes! Definitely get him into hospital. He should be getting better care than that I'm sure.

All ik is thats elder abuse and malpractice sue them cuz thats wrong