Lizzie is one of Lina's best friends, but she's so flaky that she's absolutely no help in figuring out the message from the Builders that's got Lina all excited. Doon, on the other hand, reacts instantly and positively, and he sticks with Lina through the end, when they have to put their discovery into action. We imagine Lizzie would've gotten bored partway through the investigation and decided not to stick it out after all.
We love Granny, and we're sure Granny loves Lina and Poppy, but her slipping memory means that neither kid is getting the care they need (which is a big deal in Poppy's case, since she's a toddler who needs almost-constant supervision). When Mrs. Murdo starts looking after the kids, on the other hand, everyone's fed and given vitamins and cleaned up on a regular basis. It's not that Mrs. Murdo is more loving or caring (though she is certainly those things), it's that she's she more responsible and has a better memory than Granny.