Not all men are like your "dear" H.
For several years, DH and I truly shared the house/work commitments (we both worked 3 days a week during a time where childcare was a bit dodgey). Now that we're both working 5 days a week, we share it properly.
And I'm not talking about the hoovering, dusting, cooking.
I'm talking about that plus all the other little things that need doing. Remembering to tax the car. Making DD's braces appointments - and taking her. Wiping down the kitchen sides from breakfast. Writing a note about a homework problem DS had (and explaining that we've helped him a bit).
That is not anything exceptional. I do it; DH should too. They're his kids just as much as they are mine.
It sounds like your DH is just another child, with a pretty shit attitude and no respect for you.
Don't believe his attempt to explain his being a shit husband and a shit dad with "that's how men are". No, that's how some men are.
I'm worth more than that - you are too.
In this house, I've raised my daughter, and son, to know they are worth just as much as their future family members. I don't want my DD to enter into a relationship where she is treated terribly; I don't want my DS to think it's acceptable for him to treat women with any less respect than they deserve.
You are showing your children a terrible example by sticking with this lout.