Five kids is an edge case. Besides, it's impossible to shuttle five kids to different places simultaneously by car without five parents. Not so with good public transit.
I feel that with 5 kids special provisions and certain difficulty will be necessary no matter what. You would be moving around with all children a lot less in any case. But I don't see why it's impossible to go car free; specially considering some kids will be older than others (not 5 toddlers simultaneously).