You have a couple of issues which cause the animation to be in error and so be ignored.
- There must be exactly as many keyTimes entries as values.
- There must be one fewer keySplines entry than keyTimes entries
You aren't following these rules and so the animation is in error. Here's one way to correct things...
<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 0 400 400" width="100%" height="100%">
<path d="M100 200 c0,-100 200,-100 200,0" stroke="#000000" stroke-width="4" fill="none">
<animate
dur="2"
repeatCount="indefinite"
attributeName="d"
from="M100 200 c0,-100 200,-100 200,0"
to="M100 200 c0,-100 200,-100 200,0"
values="M100 200 c50,-100 250,-100 200,0; M100 200 c-50,-100 150,-100 200,0; M100 200 c50,-100 250,-100 200,0">
</animate>
</path>
<path d="M100 200 c0,-100 200,-100 200,0" stroke="#FF0000" stroke-width="4" fill="none">
<animate
dur="2"
repeatCount="indefinite"
attributeName="d"
from="M100 200 c0,-100 200,-100 200,0"
to="M100 200 c0,-100 200,-100 200,0"
values="M100 200 c50,-100 250,-100 200,0; M100 200 c-50,-100 150,-100 200,0; M100 200 c50,-100 250,-100 200,0"
keyTimes="0; 0.5; 1"
keySplines="0.25 0 0.75 1; 0.25 0 0.75 1"
calcMode="spline">
</animate>
</path>
</svg>