How do I add window stools to my new windows?
Looks like you could "float" them level with the existing frame, but screw in from the window side (through the face away from the room, into the part you call a stool that I'd call a sill) which would firmly attach the stool to the windowframe so it could not pull away from the wall.

They are probably designed not to have one, yes - "picture-framing" windows is one more place to cut costs, and cats and plants don't have effective lobbying groups to prevent it.