You mention flexibility in the context of yoga or dance and immediately most people envision some contortionist type person doing inexplicable types of movements with their limbs. While it might be true that the physical flexibility developed in both these practices/sports are something to behold- they require a subtle and profound precision of movement.
There is, however, a second and arguably more important flexibility, and that is flexibility of the mind. Flexibility in this respect is a lot about flexible thinking. Believe it or not, there are many ways to 'get into' many of the advanced (contortionist) poses that wow, mentally. Exactly which muscles to engage, which to stretch and how much of it is a 'dance' are things that are discovered through many hours of practice. This is the reason why there is no short cut to improvement. It takes many hours of repetition (and error) to "feel" the sweet spot.
Applied to the bigger picture, flexibility of thinking and the understanding that there are many ways to achieve the same end goal is perhaps the foundation of 'wisdom'. Mind/body sports help the best of us learn to find our 'sweet spot' by being in the moment and feeling exactly when to engage and how. And hey, you may end up being able to do the splits too.