I agree that Alfred is likely the most responsable for mundane things.
If "making sure he brushed his teeth before bed, or did his homework, or picked him up from school or any of the mundane things that parents are responsible" is neccesary for Bruce being a father figure, I really doubt Bruce is father for other Robins apart from Damian (and I'm not so sure he makes all these things for Damian).
Curiously, in the current continuity, Barbara as Batgirl (before Killing Joke) doesn't seem so independent from Batman.