Having three seats gives the family more room but does not really provide a seat for the infant. There are very few under two year olds who can or will sit on their own in a seat and invariably they will spend most of the flight on laps of either parent or jumping from one to another.

By the way how is the OP to know this was a lap child and that the middle seat was not paid for? For an older child it would not be unusual for the child to be playing with parents till door shuts and then be placed in his seat for take off.

I suspect this child was never a lap child because if they truly are then they MUST be in the lap of a parent for take off (for safety reasons) or in a car seat if not on a lap. The FAs would know if the child was a lap child and would have asked that the child be in a lap (plus minus lap belt depending on airline and crew) for take off.