- they would prefer that the children be taken care of by the spouse versus a 3rd party
- if the cost of childcare were to equal or exceed the full income of the lesser-paid parent, it would be preferable for that parent to spend their time taking care of the children and removing the cost of childcare in the process
- neither parent would be able to work part-time in their current job positions, which would address your question of split duty
My wife liked parts of her job and hated others (PhD in neurobiology, working in a non-lab part of her field), but overall didn't love it and certainly didn't prefer that over raising our children. I (90+%) love what I do and we're fortunate that it pays enough for us to live on just "my" compensation. In theory, her job could have become part-time; in practice, that would have worked very, very poorly at her company. My job couldn't be done part-time or job-shared in any practical way.
So, I happen to be the stereotypical working dad, but it's not because of gender biases or pre-conceived notions, but rather because of math/finance/satisfaction concerns. We could live OK in Cambridge on just "my" comp but couldn't on just "hers"; I actively like my job and she didn't. We do 100% pool the money and there is no sense of "yours, mine, and ours", only "ours". I actively take pains to ensure that, but the tax brackets[1] and cost of daycare in Cambridge being what they are, the math made it a near even-money proposition.
So, in our case it was a combination of my wife's pay being lower and also the [dis-]satisfaction with spending ~1/4 of her life (and ~1/2 of her awake life) that way made it a near no-brainer.
1 - In essence, when considering forgoing her income, her income is effectively entirely taxed at our highest federal bracket, plus she's still paying Medicare, FICA, and state income tax on top of that. It means that about half of her "headline" pay is going to the government. That's a pretty stiff headwind.