need to animate curve control points

What the title says. I can’t insert a keyframe for them. What can be done?

Go to preferences->addons tab and look for AnimAll. Also consider just using shapekeys and animating the slider.

Curve points and handles have to be animated indirectly with a HOOK modifier. One hook modifier for each point or group of points you want to animate, target the mod to an empty or a bone (or use ctrl H) and animate those.