So the answer to the first part of your question ("What's the purpose of paragraph first-line indentation ") is that the purpose is to visually highlight the logical structure of the text (division into paragraphs).
Main point: There are several ways to reflect this logical structure visually. Examples for such logical units are parts, chapters, sections,, paragraphs, and sentences. Therefore, a text is divided into several logical units. Normally, a text reflects the structure of the idea/content/thoughts that is described. Instead use a package such as parskip or a class that sets up the style you prefer automatically. If you do prefer a different style, LaTeX actually makes it easy for you to adjust your documents to any style and there is no need to go \noindent on every paragraph as you might have feared from the description on overleaf (that just describes how to turn it off for individual paragraphs which is sometimes useful. That's about the background.īut there is no reason to "struggle" with it. To be able to nevertheless always enable the reader to see that a new paragraph has started the first line therefore typically indented by a small amount (and since after a heading this visual clue is unnecessary it is normally left out). If you do this, then you have the issue that the last line of a paragraph can become completely filled once in a while. The most common approach in "western" typography (which is where TeX originate) is to typeset text justified and (last not least to save space) not to put extra white space between paragraphs. 
Perhaps your background is CJK scripts in which case the situation is probably different.
There are publications that indicate paragraphs by leaving some extra space between them (most often with ragged right typesetting) but I would claim that they are in the minority if you go into a library and open books at random, so I'm a bit surprised you state you have never encountered them in "real" life".
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