I can imagine situations where if an employer wants an employee bad enough, they would take the employee and work around that employee’s scheduling needs. Same thing would go for an established employee who the employer wants to keep and would adjust to the employee’s scheduling needs. Where I work (for the government) my hours are “office hours” so it’s predictable and therefore, my child care needs are predictable and this is not an issue. Also, our sick time is for illness of the employee and also for illness of a family member (meaning not just your child but also your parents, etc.) so it’s pretty liberal in that sense.
If I worked in the service industry (hotel, retail, customer service, etc.) I would have more of a problem. A friend of mine works for a hotel and she is older and kids are grown, so she works weekends, etc. New employees at that hotel have a “deal” where they have told the boss they’re not available on weekends because of child care issues. The hotel is desperate enough for good employees that they accomodate the new employees scheduling needs. This makes it tough on my friend, but the hotel is in need of help and so they deal with it best they can until they can hire and train more staff.