Which means that other people who do go to work support those who sit around and do nothing. As someone who works, I have no desire to support people who don't work because they don't have to.
> people can choose whether they work or stay home raising a family, and nobody _has_ to work to make ends meet.
Parents who leave paid employment to raise children are far from "sit[ing] around and do[ing] nothing". If we value future conditions (even restricting that calculation only to economic productivity, though I'd argue other considerations also matter) then investing resources in raising and educating children is rational, even for those of us who don't have them ourselves.
I'm the breadwinner for someone who stays at home and she's immensely valuable. A fact I'm quite aware of as she just left for a few weeks on a trip and now I'm working and doing what she does. She does all the house chores, picks up medications/runs errands, is trying to get a business of her own off the ground (is making ~500/mo with potential to scale if she's allowed time to actually do it), etc. She also contributes a lot to making sure our household is embedded in our community, which in turn enables us to both more efficiently help others (when we can) and to receive help when we need it without relying more on the government/taxpayers. She does the work required to do things like help our parents when needed (we're siblings), which would otherwise be paid caregiving paid by, you guessed it, the taxpayers! She also makes me a way better worker because I'm the one person on my team who actually can focus on work all day and doesn't have to frequently step out for life things.
And this is without children in the home.
Not to mention students, people with disabilities (some of whom may end up being a greater boon to society if they're allowed to reskill or be pickier about jobs rather than being forced to work themselves to the bone until they can contribute nothing + their medical needs end up being worse than they would have been if they were accommodated in the beginning), etc.
The idea that the only way people can contribute to society is via paid employment is a sign of a lack of creativity.
The idea that socialism or whatever government system would mean that nobody has to do any work is only appealing to downwardly-mobile elite children, which America has a lot of, and who mainly like to live in Brooklyn, start political podcasts and do drugs.
People really actually have to grow food and clean the bathroom - the point of economic systems is to get them to do it, not to get them to not do it.
That said, in my country (Netherlands), a lot of service industry jobs (waiting, dishwashing, store restocking, etc) are jobs that people have temporarily or part-time during their studies. But it's a crooked comparison as tipping culture is still very much optional over here and staff gets paid more.
As for your update, in a free capitalist economy, no, you don't need to pay Carol more than the others, you pay her a fair wage according to laws, hours paid, etc; if that is not enough for her to cover the cost of living + raising kids, the government needs to compensate. As an employer you can support her private life by offering flexible hours, of course.
That said, in an idealized, socialist country, people can choose whether they work or stay home raising a family, and nobody has to work to make ends meet. Unfortunately, this is often not compatible with capitalism or government policies / expenditure.