Working from home and having a home-based business are not the same.
I work from home. I have a full time job, and an office. I travel. I work OT. I don’t have kids, but if I did the expectation would be that they are in childcare.
There are some work from home jobs that are legitimate and very flexible, but again, during the hours you are working you would be expected to have a quiet, business environment and children supervised by others. There are call centers that employ home-based agens (1-800-flowers.com, Jet Blue, etc). Some let you work at home but require you to live within X miles of one of their offices, others are completely virtual.
Consulting work can also be done virtually if you are in the right field, like computer programming, training development, technical writing, etc.
Contrasting that is a home-based business. You set your own hours. You create a product or service, sell it IRL or online, or do some sort of work that can be done when kids are sleeping, etc. Most of these don’t produce the type of income that a full time job does-- work at home or not.
I have a couple of friends who are FT moms and have home-based businesses. They work hard, but do have flexibility. One sells on the internet. She makes a product and sells online-- she now has some wholesale customers. She is very busy, but I don’t know how much she actually makes… I do know she’s been in business over 5 years.
The other had an in-home bridal accessory business she had for about 6 years. She doesn’t do it anymore, but she had the spare bedroom set up as her bride room. She did tradeshows (like the Bridal Extraveganza that comes to major cities) she advertized, she got word-of-mouth referrals. She ended up giving it up when she had #2 b/c it was too hard to have people coming to the house and the kids needing her attention. She made “spending money” not a second income.
Oh, and another makes hand-made jewelry and sells online and at craft fairs. Again, I don’t know how much she makes, or if this is just “spending money” and something she enjoys doing.