"Maybe target asking H to help with one teeny new thing per week?"
I'd suggest a different tactic. You really don't need to coddle H to get him to act like a big boy.
Instead, treat H immediately as a man who is competent in all things. Maintain that assumption in general. Let him be the one to prove otherwise in any particular case and then address that if it ever becomes necessary.
BTW, the kind of hypervigilance you are describing is incredibly exhausting and draining to be around. It can also be very selfish -- whether you want it to or not, it winds up making almost everything about YOU in some way. H not feeling well and acting grumpy, for instance, becomes about YOUR anxiety and then YOUR relief.
I know you recognize this, I am just reaffirming the importance of trying to shift away from that hypervigilant mode -- focus more on YOU and getting what YOU need than on what is going on with others all the time. Ultimately, it is less selfish and more giving of your authentic self.
