Dad won't let anyone else care for him and his full-time caregiver needs a break. How can we convince him?

I can tell you how I handled a similar experience. I am the Caregiver to a woman that is the same way. I used to work 168 hours a week. It was exhausting, and nearly put me in the hospital due to my employer being of the demanding type. She refused to allow any other caregivers help her. Wouldn't even give them a chance! If they didnt quit, she would fire them within hours of beginning their shift. I had to be proactive in the resolution if it was going to get any better, what I did was talk to my boss about it. I asked her if she thought I had a pretty good idea of what a good caregiver should do, and what type of person could fill this space. She agreed that I did. I then asked her for permission to hire another caregiver myself. She agreed. I then posted an ad, checked references, and we interviewed people together. We talked about who the successful candidates were, and with a little encouragement, she made the final decision and choice herself. I hope that helps.

can you threaten that he's either nice or he goes to NH for 2 weeks while she is out of town? Threaten pizza delivery for 2 weeks & you calling once a day to see if he is still alive? Make sure that he knows that NO ONE is trying to replace her with another ("inferior") caregiver, but the entire universe does not revolve around him either. Basically he is spoiled, and maybe afraid that she will leave him.
Find the firmest male nurse that won't put up with his BS, and just tell him that you don't EXPECT him to be happy, but for 2 weeks a year he can be civil and just deal with it (then of course as long as the caregiver will put up with the abuse you ignore the tantrums). Is there a reason he can't be alone for any period of time? Would a cash penalty if the caregiver leaves deter him--ie if the pre-arranged one leaves because he's mean he has to pay 2x as much to hire one from an agency on an emergency basis...I'm sure you can find moe expensive options to threaten him with. Sometimes money is a heck of a motivator.

IMHO she's basically his "wife" --no one stays 29 years for a paycheck!
What does she say? Does she have any relatives (niece, etc.) that could maybe be options (so that he is "helping" her relative). She has to be truely onboard with it too, of course. Maybe a Dr.s note for her that she needs a day off per week for her health (she can't be under 50, right?).
Has anyone made sure that this saint will be cared for after your Dad dies? She's basically spent her entire career caring for him.

First, you need to tell the caregiver to grow a backbone. 7 days a week for 30 years? That is just insane. She needs to stop catering to him and demand time off. And tell him if he won't do that, she is leaving. I also question whether she is one of those "people who need to be needed," because no normal person would keep a job like that. And I don't know if it will get through to him, but have you talked to him about how incredibly SELFISH he is being? Everyone needs time off to recharge. That is very unfair to her. My Mom loves the caregiver we have 2x/week. But I also wanted her for Saturdays. She has a life, and didn't want to work Saturdays. I didn't even tell my Mom, I just told the agency to send someone. If she likes her, good. If she doesn't, I would find out why and see if there is a solution. I've stopped asking permission. I have her POA and I need help. My brother refuses to, so I hired a caregiver last year. We have lost 3-4 (due to different reasons -- 2 of them were really odd), but now it is getting better.