Child safety (which is what we are discussing) is not a "full service option"; I dare say its not even a 'service' but a 'right'. I bet every LCC will say somewhere "your safety is our first consideration".
Whether or not the parent has paid for seating, or doesn't know which way is up, if the airline accepts the child to fly, it needs to be cognisant of its particular safety needs.
No-one's seat is unchangeable. We can all get punted for various 'operational' reasons. I wouldn't like it happening to me, but I see no issue in a FA moving a 'paying' or 'organised' pax to put a minor next to one of the adults in their party if somehow the minor has been split off. (Again, note, not having the family sit together). It should have been sorted beforehand, but if not, that's what's got to happen. Its a much better reason for shifting me than just so some CL knob can have the seat I've selected.