Percentages vary depending on the source / methodology, but anywhere from 61% to 97% of people of either gender self-report a preference for staying home instead of working a full time job. Either as a full time stay at home spouse with no job or as a primary caretaker with a part-time job instead of a full-time job.

For women we have these sources, which report rates from 61% to 83%:

And for men we have these sources, which report rates from 68% to 97%:

For some reason this isn't obvious to most people. With the push to get more women into the workforce, along with the rhetoric of "unpaid labor", people assume most women want full time jobs and hate taking care of children.

And the idea that men prefer working instead of taking care of children is just assumed to be true because of how we view gender roles.

But what if I told you that there was no gender difference here, and that most people do in fact prefer taking care of children and doing household chores over working a job?

Moreover, what if I told you that this gender norm is primarily enforced by women, not by men, or the patriarchy?

Sources for that:

Women actively look for men who make more money. Married men make more money not because the wife is cooking and cleaning which somehow helps them earn more, but because their wives saw that they were on promising career trajectories before marrying them. The incentive for men is therefore to earn more money to attract a partner:

Ludwig, V., & Brüderl, J. (2018). Is there a male marital wage premium? New evidence from the United States. American Sociological Review, 83(4), 744-770.

https://journals.sagepub.com/doi/abs/10.1177/0003122418784909?journalCode=asra&

This study shows that a lack of men will cause women to focus on their careers instead of finding a husband. The effect is weaker for women who are more attractive and therefore have a better chance of finding a high earning husband. This indicates that "husband hunting" comes first and "career hunting" comes second, as the obviously less less desirable option:

Durante, K. M., Griskevicius, V., Simpson, J. A., Cantú, S. M., & Tybur, J. M. (2012). Sex ratio and women's career choice: Does a scarcity of men lead women to choose briefcase over baby?. Journal of personality and social psychology, 103(1), 121.

https://www.ncbi.nlm.nih.gov/pubmed/22468947

A man's wealth is directly correlated with how women rate him. A similar effect the other way around is small to non existent. Men select for looks and personality. Women select for looks and money. Thus putting more pressure on men to earn money compared to women:

Wang, G., Cao, M., Sauciuvenaite, J., Bissland, R., Hacker, M., Hambly, C., ... & Speakman, J. R. (2018). Different impacts of resources on opposite sex ratings of physical attractiveness by males and females. Evolution and Human Behavior, 39(2), 220-225.

https://www.sciencedirect.com/science/article/abs/pii/S109051381730315X

A loss of income is the single largest predictive factor for a wife wanting to divorce her husband. The same is not true the other way around. And interestingly, a wife's inability or refusal to "cook and clean" does not effect a husband's willingness to divorce her, either. Indicating that neither side of this is enforced by men, even indirectly.

I'm not sure if this study goes into this, but my guess is personality issues, things like nagging and harassment, might be the biggest factor for wanting a divorce for husbands:

Killewalda, A. (2016). Money, Work, and Marital Stability: Assessing Change in the Gendered Determinants of Divorce. American Sociological Review, 81(4), 696-719.

https://scholar.harvard.edu/files/akillewald/files/money_work_and_marital_stability.pdf

All of these trends start long before people get married or have kids, so children aren't responsible in those cases.

However, when children do enter the picture, it's women who actively enforce these gender norms. Most men would love to stay home and help take care of the kids, but mothers send a clear message to fathers: "I'm taking care of the kids, in fact I'm naturally better at it than you, so you need to give up on that and instead go out and earn more money to pay for everything".

Steward, Melissa. (2017). Maternal Gatekeeping & Why It Matters for Children. The Father Factor.

https://www.fatherhood.org/fatherhood/maternal-gatekeeping-why-it-matters-for-children

A few other sources that might be interesting...

Public views on men having to provide for their family:

https://www.pewsocialtrends.org/2014/09/24/chapter-1-public-views-on-marriage

Social stigmas in the dating market for unemployed and underemployed men:

https://www.prnewswire.com/news-releases/social-stigma-for-unemployed-men-75-of-women-are-unlikely-to-date-them-160378575.html

Whether you think it's on purpose, or the byproduct of biology (ie the fact that women are the ones who get pregnant and breastfeed), this gender role does not appear to be enforced in a top down manner by men or because of some kind of patriarchal conspiracy among men.

It also benefits women more than it does men, which some people might even call a privilege. This effect leads to:

  • A better work-life balance,

  • higher reported levels of life satisfaction,

  • higher reported levels of happiness,

  • more free time,

  • more time spent at home,

  • better health outcomes,

  • lower mortality rates,

  • and a longer life.

So if men are responsible for this gender role, it is under misguided pretenses.

The traditional explanation that women have less time for a career because of unpaid labor may play a part, especially as a kind of feedback loop, but this effect exists long before marriage and children enter the picture. The answer to the question of who makes less and should therefore spend more time at home with the children is already decided long before the question gets asked. In part because married men have already spent most of their lives sacrificing their time, health, and happiness in order to maximize their income to attract a wife.

It's also women who tend to "decide" who does what in a relationship after children are born. In fact there's a plethora of research showing that wives and girlfriends command significantly more power and control in relationships than men do. This isn't meant to point fingers and blame women for anything, it just counters the common narrative that we hear about this (where men are by default the ones blamed), so I figure it's worth mentioning. I'll leave those studies out for the sake of brevity (and for veering outside the scope of the OP) but I can post them if requested.