Large parts of our family life has been spent car-free. When the kids were small, the older one went on the crossbar, the smaller one on the seat in the back and the weeks shopping in the panniers. When I say my wife is a tower of strength I am not speaking metaphorically . . . and revealing how much of the grunt work I left to her <blush> And we live in a hilly area. A very strong non-sus mountain bike with gears suitable for caravan towing was the answer.
When the kids got bigger they seemed reluctant to use their bikes much, and we slid back into motoring habits. Should have kept the money back and spent it on a fleet of tandems/trailerbikes/cargobikes etc. Hadn't seen the light then. Ironically both are now happy adult cyclists with very nice bikes bought for a song - ironic.
Then I discovered the joy of folding. No more smelly car-commutes thank you very much. Out by train, back by cycle route.
Nowadays wife is still driving because her job involves multiple visits carrying equipment on tight schedules. (I still think a cargo bike would be better
). We also use it for ferrying our aged mums about (but wouldn't a bicycle-rickshaw be so much more stylish?
). I am pretty much car-free however. My bike is my car and it does the lot: shopping, business trips, commuting etc. Somebody once said 'Yes but you can't carry a length of four-by two on a bike'. As my (recumbent) bike basically IS a length of four by two, I don't see the problem . . .
Lately I am getting even more hardcore and organising my schedule so I can just walk instead of lugging a heavy folder on the train. Not always possible, but whatever transport you use, if you regard the traveling time as time you don't have to spend in the gym because its your exercise, then it doesn't seem like time wasted. This last point I think is the key to 'losing motor anxiety' once you have got over the status issue. Especially when you factor in the time wasted on many journeys wasted as you get diverted around one-way systems desperately looking for somewhere to park the thing.
Finally, consider the car-club route, if there's one in your area. More flexible than standard car rental.
Go for it.