No, feature engineering is not an important step for deep learning (EDIT: compared to other techniques) provided that you have enough data. If your dataset is big enough (which varies from task to task), you can perform what is called an end-to-end learning.
To further clarify, according to this article, deep neural nets trained with backpropagation algorithm are basically doing an automated feature engineering.
I feel like with enough regularization, the deep learning models don't need feature engineering compared to some machine learning models (SVMs, random forests, etc.)
That is basically correct. Beware, you need a large dataset. When a large dataset is not available, you will do some manual work (feature engineering).
Nevertheless, it is always a good idea to look at your data first!
I would also like to quote Rich Sutton here:
We want AI agents that can discover like we can, not which contain
what we have discovered. Building in our discoveries only makes it
harder to see how the discovering process can be done.
Perhaps this statement is more true with Deep Learning than with previous techniques, but we are not quite there yet. And as user nbro rightfully pointed out in the comments below, you may still need to normalise your data, pre-process it, remove outliers, etc. Thus in practice, you may still need to transform your data to a certain degree, depending on many factors.