A parent travelling with two children under the age of 2 needs to plan carefully. It appears that on one flight she took along another family member who took charge of one infant or young child and obviously that was sensible and mitigated any problem. On the second flight she had attempted to solve the problem by utilizing a car seat that unfortunetaly wouldn't fit in the seat.
Although another passsenger offered to assist by taking charge of one of the children, the airline it seems wouldn't permit that to occur.
I can imagine that staff might have (as sometimes happens in these difficult situations), quoted acts such as the Child Protection Act to excuse an action, when in reality they probably have no idea what the act actually states or encompasses ? However parental authority it seems was being given to another adult to act in loco parentis on this occaision thereby giving the necessary permission.
The problem arises because such permission it seems, is contrary to the airlines rules (which presumably are stated in its booking terms and conditions). If that is the case then the passenger implicitly accepted those conditions when booking the flight. The airlines staff were obliged to ensure those rules were executed, and as unfortunate as it was the passengers had to be removed.
It sounds like better planning by all, would have prevented this situation being allowed to develop all the way through to the "onboard stage". I have always believed that child seats (where required ) should only be supplied by the airline as a part of the advance booking procedure. Children under 12 should never be seperated on board from the responsible adult or guardian unless alternative arrangements have been made with the carrier. No more than one infant or child under 2 should be allowed to accompany one responsible adult.
It seems that neither the airline nor its handling agents did a very good job of allowing this situation to develop. However with the best will in the world we all know these type of things sometimes happen. The crew were the last link in the chain and shouldn't be blamed for then having to be the ones who ensured the compliance.