OEBB (Austrian Federal Railways, the state train operator) does offer a family railcard ("Vorteilscard Family"), and it's hardly expensive:
"The Vorteilscard Family offers you a particularly inexpensive option to travel together with children. For only €19 a year, up to 4 children under the age of 15 can accompany you for free."
But when I recently planned a trip from Linz to Vienna for our family it would have cost around 120€ for the train (not including tickets for bus or subway to get to/from train station). Driving costs around 40€ for gas (not including parking fees).
It's 184 km one way. German tax authorities assume 30 cents per km (~110 EUR for a round trip), Austrian tax authorities use 42 (~154 EUR for a round trip). Now, the actual marginal cost may be lower if you already have the car, but it's far from obvious once those hidden costs are considered. Especially as you have to actively drive for those 4 hours instead of reading a book, watching a movie, ...
Every sunday hundreds of people drive from all over Austria to visit the zoo in Schönbrunn. They take the car because the train is way too expensive. If taking the train was cheaper then some of those people would take the train.
Yes, for each of those families it's just one trip. But in aggregate it's also a lot of people.
Why is it more important to get one person to use public transport for 500 trips rather than getting 500 people to use public transport for one trip?