I don't have time to deal with it this evening. It is time for bed. However, you have designed a set of tasks that are not pertinent to the issue. Squares are not the problem. Curves are the problem.
If I moved a picture in picture across the screen using sharp right angles, you would scream that it was not organic looking. And you would be right. Bezier curves are tough enough with two keyframes let alone three or more.
Design a task where the square (or rectangle in this case) moves from the lower left hand corner to the lower right hand corner passing through the top middle. And do it with three keyframes. Then try it with more than three. I think the technical term for that is "oops". You will immediately want to delete the extra keyframe.
Then for a second project, take the square from the middle left to the middle right in the shape of a sine wave using the minimum number of keyframe. Nice and easy. How many did it take to go down then up then back down? I can't provide a picture without giving away the answer.
Others may comment, but I think you might be asking the wrong questions. I think it is great that you are asking but until you get one of those damn wobbles and find yourself erasing all the keyframes and starting over, you won't quite grok it all. Your best bet is to practice with the keyframes. Learn them. They are your friend. Then, if you still think they should all turn on automatically, we can discuss what it means to look organic and how the lengths of the handles change the bezier curves and the difference between continuous and other types of curves.
Just don't use straight lines - who does that? Oh, sure, sometimes.
Oh, then start using keyframes temporally as well as spacially. Then it gets interesting. Ease in and Ease out. Anything in between is in the way. Use my first example moving up the hill slower and slower as though getting tired, then pick up speed faster and faster as you slide down the hill. Don't forget to pause at the top of the hill. That's organic.
Let me know what you think of my answer. If you don't like it, please tell me why. And if you have questions, feel free to ask. I have a day job so please don't expect immediate answers.