Okay, if we have "functional strength", then what is "non-functional strength"? Makes no sense, does it? All strength is useful.
Lifting heavy weights
will give you "usable power". If you can do 10 chins with a 20kg plate on you quite simply you are going to have an easier time climbing rocks than if you cannot do even a single chinup. If you can deadlift twice your bodyweight then you are going to be able to really smash people in rugby, and moving your friend's fridge when you're helping them move house will be a breeze, and if those deadlifts were with bare hands then it won't be your grip failing on those climbs. And so on.
But that is not the same as control, such as gymnasts have. Control is essentially agility; agility and strength are related but different.
That said, I have heard of gymnasts who went on to be weightlifters (not powerlifters), but I don't know of weightlifters who became gymnasts. Perhaps it's easier to develop control first and raw strength afterwards than vice versa? I don't know.
One of my teachers at PT school is a PT (surprise!), and says he's got a lot more into bodyweight work recently because many of his clients travel frequently, so they don't have equipment available.
I think bodyweight work is an excellent way to begin for an untrained and unfit person, it gives them a slow start, some confidence, and lets them develop the
habit of exercise. If someone has the self-discipline to do pushups etc for three months at home, then they can be fairly sure that if they sign up for 12 months at a gym, they'll use that membership, not bail after a month or so.
Bodyweight work also looks cool and impresses people a lot. I think that is really what Keen Katie and Josh are responding to. If I can do ten handstand pressups I can show that at a party, benching 100kg is harder to demonstrate