Sometimes, my kids will say they're bored. My response to them is that my job is not to entertain them. They totally get it, though sometimes they're still bored...
When I was a kid, I used to just watch TV and wait for friends to come at home to play.
In one particular afternoon, there was nothing good showing on TV and my friends are away. I complained to my mom that I was bored(implying that she should solve this "problem" for me).
My mom basically said just what you posted above. That it was my responsibility to find my own fun.
That was an epiphany that had a tremendous impact in my life.
A parent's job is to give their kids opportunity to learn and grow. You can't always force it on them, and they won't always be interested.
You can give them them the opportunity to do something that will help them grow (learn, build character, etc) or they can be bored, but if they choose to be bored, they need to be told that they have options, and they are the ones choosing to be bored.
Think about you adult life: When are the moments you have some kind of epiphany about something that currently bothers you? For many people this happens when out for a walk, under the shower or a similar situation when the own mind is free to wander.
Cognitive development is important, but school, discourse with parents etc. is usually enough for the healthy development of most kids. For a successful life as an adult, the emotional development is just as important, though.