Sounds like you are referring to 'mass transit' ie: commuter trains. Not 'public transit' ie: busses
Public transit is routinely used for ferrying children and groceries. In fact, in many municipalities, that is its majority use. Additionally, there is a significant stigma associated with its usage in many areas. I haven't used it personally since I was a teenager in Denver, but judging by those whom I see waiting at bus stops, the patterns haven't changed much in 30 years. People who cannot afford cars use the bus. Often with carts full of groceries and strollers filled with children. Add in non-optimal transit patterns and scheduling issues, public transit is viewed by most as the transit choice of last-resort, only to be used by those unfortunates who have no other choice.
I use mass-transit daily to commute via train to Manhattan for work. Commuter trains mostly solve this issue fairly well, although there are still issues around scheduling and quality of service (ie: my 90 minute commute is only a 90 minute commute if I am able to leave work at exactly the right time. Get held up for 5 minutes on my way out of the office and I get to sit at Grand Central Terminal for 30 minutes waiting for the next train, turning it into a 120 minute commute).
In major cities (Subway in Manhattan, The Underground in London, Tokyo's subway system, etc) there is a hybrid solution that actually works well, being used both for mass transit and public transit. These options generally have minimal delays (5 - 10 minutes between trains, max) and service all types of commuters. Of course, these types of systems are only feasible in areas with sufficient density of population and commercial interests and should really be viewed as exceptions rather than a model to be adopted by all regions.