If you have feet, don't take them for granted.
We use them without thought most of the time. But remember, a foot injury can change your world.
The discussion of power-force generation for speed and agility is a good one but tries to truncate one body part use without the other. Here's an interesting Q&A link about "Quick Feet and Agility Drills". As difficult as it is to think outside the box, with the popularity of Pilates infiltrating the dance, fitness and sports performance domains, the box is slowly changing shape into a global training perspective. In 21st century training, squaring the circle is not optimal.
This "Bulgarian" lunge is remniscent of Joe Pilates style Russian Split. No we don't do Pilates...
We see how the sports industry is evolving. The modern dance world is changing shape too. But traditional Ballet continues to cripple their dancers for the sake of something that can never be questioned--tradition. But when and how does tradition take hold? Can it evolve?
One message the dance world gives to the sports world is this: "Get all of your body weight out of your heels!" And, sports training weighting into the heels adds balance to that message.
However, if you want speed and agility, you're not going anywhere fast if all of your training and stances are coached by pressing all your body weight into the heels. The arches of the feet help distribute weight through the bones of the feet, and the muscles of the feet got to be resilient.
So yes, you need your feet for speed and agility. Now is the best time to strengthen and train with an all-encompassing understanding of their power.