So would your ideal situation be that DH always gets up at 5?
That seems reasonable to me, so long as you are doing all the night wakings. There has to be a time when you feel fully "off the clock" and know that he's got it, and if that's 5-7am so be it. I'd do the following:
weekdays - you do nights and he does 5am start
weekends, you alternate, or if he simply won't hear them crying in the night and you have to get up, he either sleeps nearer them/uses a monitor, or if there's no way you can sleep through, he then has the 5am start both days.
I might be biased as DH and I both work, our children are older, and to have time for a full day's work and do the school runs and other things, I will always be up at 5 and he will always be up at 6. Obviously at the weekend we can sleep and take it in turns and we don't have night waking any more. But the idea that a 5am start is a hardship for someone who just goes to work seems a bit pathetic.
I also have no sympathy for people who "just don't hear" or don't wake up, to me that signals they aren't fundamentally and subconsciously feeling responsible. What would they do if they were on their own at night?
My remarks might be different if he's a brain surgeon or running an international consultancy and has 15 hour intense shifts, and you get a 2 hour nap time and work 3 mornings a week, or something...but somehow I don't think that's the case here.