You mixed in a bunch of three different things. Jade is a template engine for a node. sass and less are generators for css (or as they are called fashionably, preprocessors), and emmet is a plug-in for the editor. jade and emmet are not preprocessors.
Why did preprocessors appear? everything is simple. Modern browsers are developing fast and new features appear frequently. But not always these opportunities appear in all browsers at the same time and equally. Preprocessors just allow you to hide these features.
Once upon a time, it was very difficult to make a table with lines one pixel wide and with rounded edges. Especially a lot of problems were in IE5.5 (yes, I also saw IE3 / IE4). In the modern world with preprocessors, this task would be solved simply - for the preprocessor they would create the necessary style, and the preprocessor for the necessary set of browsers would generate the correct css.
Or the second task - for example, I want one element to be 10 pixels wider than the other, but if it is more than 100, then by 20. On the classic css, this, as far as I know, cannot be solved. It is necessary to use js. Less can compute arithmetic expressions when compiling and do everything on its own.
Can Emmet help everyone in this? not. He helps edit. But if tomorrow there will be another browser with support for HTML100500, where there will be a bunch of new features, sass / less will add plugins and will generate css for them. The creators of the jade template will go somewhere else as well. And emmet users will have to edit everything manually.