You can indeed fit a polynomial to your labelled data, which is known as polynomial regression (which can e.g. be done with the function
numpy.polyfit). One apparent limitation of polynomial regression is that, in practice, you need to assume that your data follows some specific polynomial of some degree $n$, i.e. you assume that your data has the form of the polynomial that you choose, which may not be true.
When you use a neural network to solve a classification or regression problem, you also need to choose the activation functions, the number of neurons, how they are connected, etc., so you also need to limit the number and type of functions that you can learn with neural networks, i.e. the hypothesis space.
Now, it is not necessarily a bad thing to limit the hypothesis space. In fact, learning is generally an ill-posed problem, i.e. in simple terms, there could be multiple solutions or no solutions at all (and other problems), so, actually, you often need to limit the hypothesis space to find some useful solutions (e.g. solutions that generalise better to unseen data). Regularisations techniques are ways of constraining the learning problem, and the hypothesis space (i.e. the set of functions that your learning algorithm can choose from), and thus making the learning problem well-posed.
Neural networks are not preferred over polynomial regression because they are theoretically more powerful. In fact, both can approximate any continuous function , but these are just theoretical results, i.e. these results do not give you the magical formula to choose the most appropriate neural network or polynomial that best approximates the desired unknown function.
In practice, neural networks have been proven to effectively solve many tasks (e.g. translation of natural language, playing go or atari games, image classification, etc.), so I would say that this is the main reason they are widely studied and there is a lot of interest in them. However, neural networks typically require large datasets to approximate well the desired but unknown function, it can be computationally expensive to train or perform inference with them, and there are other limitations (see this), so neural networks are definitely not perfect tools, and there is the need to improve them to make them more efficient and useful in certain scenarios (e.g. scenarios where uncertainty estimation is required).
I am not really familiar with research on polynomial regression, but it is possible that this and other tools have been overlooked by the ML community. You may want to have a look at this paper, which states that NNs are essentially doing polynomial regression, though I have not read it, so I don't know the details about the main ideas and results in this paper.