I think he needs more on the job training. Men do not babysit their children. They parent them. If he doesn't do those things when you are around, he should start pitching in so that he knows what to do, even when he's tired. Tired? I have a cold and am exhausted, but can I abdicate my maternal duties when DD is awake and home when I am tired? No. So either he shapes up in the fatherhood department or he pays for a real babysitter, IMO. And if he doesn't like being replaced, then he needs to reconsider his behavior.
DH likes to quote Robert Heinlein: "A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying*, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.”
DH darned well changes a diaper around here. And a hundred other things. He might not do things just the way I do them with DD, but I know that generally speaking I'm leaving her in good hands when I leave her in his care. You need to be as confident with your spouse. They need a father, not a part time babysitter.