Why RNNs necessary for time series? I got it that when using time series data, I have to use a RNN



The highlight here is that the neurons receive their output again as an input, so they can take into account the previous step.

But what I do not understand is: Why cannot regular NNs do the same? For example, when I provide an order like

0, 1, 2, 3, 4, 5, 6, 7, ..


the NN should register that there is a pattern in it. If these numbers represent days or seconds, it is not important.
So why are self-looped neurons now necessary to do so?
To be clear, RNN are not necessary as there exist other options. However, RNN are useful since they implicitly deal with temporal relationships between inputs.

Regular neural networks are idempotent, meaning that the same input can be passed multiple times and the same output will be computed (obviously if the network is not updated in the meantime).

This idempotence means that regular neural nets cannot implicitly "see" temporal patterns. To do so, they would need to be provided with temporal information as you suggest. This is actually what transformers achieve! By providing temporal embeddings on top of the input, it always a regular neural net to learn from temporal information.