cost wise you're not any better or worse off taking uber from airport or HK Station.
However my family has a general tendency to take the train. Also it avoids any possible traffic on the highway (yes, usually it is exactly those taxi drivers that you are trying to avoid). At HK Station they pick up at street level outside where the in-town checkin desks are...
PS taking a taxi from HK Station: (it wont work for you with a whole family and bags but if you're 2 adults who knows what is going on w min bags, yu can get on a taxi there too, but just get on and tell driver where you wanna go). It's not as bad as how Chrisli says but i dont enjoy dealing with cabbies who loves over charging fares and luggage fees (i know how big my rollerboard is and its not chargable according to law. some cabbies think otherwise and i always tell them to measure if if they are not happy)