Create custom animation via C# script at runtime?

Seems like the simplest thing: how do I create a custom animation via C# script at runtime?

That is, I want to compute the keyframe values I want in my script, and set them at runtime.

I've looked at lots of docs and posts, but am still lost.

How do I even access an AnimationCurve at runtime? Ideally I'd just create empty curves in the editor, then set the keys at runtime. Or do I have to create whole new AnimationCurves? There's AnimationClip.SetCurve(), but the docs say this only works in legacy animations. Why?!

I just want to say "Here are the keyframes, now trigger the animation".