The parent asked regarding overfitting, because fitting a polynomial to a finite set of given data points can be viewed as an instance of a machine learning problem (inducing a regressor or classifier). What is called "control point"in the OP would be called the "training data". Now whether polynomials are a good idea for any given problem is a matter of the nature of the data, but overfitting in this context means that the polynomial induced via the parameters A, B, C and D can generalize to further (unseen) data points or not.

In the gaming context this is not a relevant question because, as you rightly say, the problem is "just to fit" what we have.

But in scenarios where the polynomial is an intensional representation for a function specified extensionally, i.e. via as a set of finite data points, and where we gradually become aware of additional data points, we can use the polynomial to forecast further data points - the interpolated ones.

Recent example paper: https://arxiv.org/pdf/1808.03216.pdf

The polynomial will go through your samples exactly, but you might have arbitrarily high error between the samples ( runges phenomenon), if your data is noisy at all it is much better to fit a low order polynomial to your data that minimises error. If your curve has no noise and you are free to choose the sample locations , minimax is the best way to guarantee minimum error between samples.