[Arthalion]

Pater familias is a confusing concept to discuss, since the actual powers granted by it varied greatly through the centuries. For part of Roman history, the fathers right to execution was constrained to casting out or exposure. This only made it an effective form of execution for very young children. Elsewhere in Roman history, that right was absolute, but was constrained by pietas. Unless a father could show that an older child absolutely deserved to be put to death (i.e., a teenage rapist or murderer), any father killing his child beyond infancy would have been considered monstrous. Pietas created a relationship of mutual aid, respect, and affection between fathers and sons (and to a lesser extent, daughters) which Roman societiety expected a pater familias to support and enforce within his household. During periods of unlimited "pater power", a pater familias who killed his children without cause would have been seen as violating one of the three Roman virtues, and would have become a social outcast as a result.

The unquestioned killing of children was only acceptable if the child did not yet have a soul, which largely restricted it to the first year of life. Admittedly, the killing of somwhat older children was also largely overlooked if the child turned out to have some sort of developmental defect.

[Lucius Cornelius Sulla]

Pietas created a relationship of mutual aid, respect, and affection between fathers and sons (and to a lesser extent, daughters)

One of the best known examples of this was Marcus Tullius Cicero's affection for his daughter, Tullia, known from his letters to her.

[Sherman Logan]

You are correct about the difficulty of discussing “Roman” institutions without further qualification. We are, after all, discussing a timespan of more than 1,000 years, almost 2,000 if you include the Byzantines, in a society which changed from a backwoods village to the unchallenged ruler of the world.