"I know it is a common opinion that single parents must work so as to teach their children about work ethics"
I don't understand how the word single is relevant in that sentence, and I also doubt most parents (single or otherwise), work to teach their children about work ethics.
Most people work to pay the bills, and/or because they enjoy it and get other things out of it like mental stimulation, companionship, personal development and sense of achievement, etc etc etc
If you prefer not to work and can afford not to, fine.
If you need to work but can't because you can't find a job, or for health reasons or similar, so have no choice but to rely on the state for your income, fine.
If you can work, but just prefer not to, can't afford that choice and then make it anyway expecting the state to pick up the tab, that doesn't sit well with me. I just can't picture myself even considering that as an option tbh.
If your children were miserable at nursery/after school club for the part time hours you were working, any reason you couldn't explore alternative childcare options?