It seems that sims will always take a taxi if they're taking an infant or toddler with them somewhere, even if there's another driver available. (This wasn't always the case - I've seen pictures of an adult sim trying to hold an infant and a steering wheel at the same time. Usually the infant ended up upside-down in the adult's lap.) You didn't mention the age of the third person in your family - is that possibly an issue?
Other than that, the way I most reliably got people to use the Motive Mobile was to put it in their inventory and have it be their preferred vehicle, and choose "go to" interactions on the Motive Mobile instead of the sim or the map a few times until they got the idea. It might help to have them walk a few steps off the lot with the Motive Mobile in their inventory, too, so they're not on the same lot as another vehicle.