Should Pinyin be written as it is (collectively) pronounced or in terms of the individual syllables?
I want to emphasize what @user3306356 says in a comment.  Hanyu Pinyin (漢語拼音) is an officially established system and has explicit rules which you can see  here. As @jdods remarks in a comment to this answer, that means tones are written as assigned and not as pronounced, while syllables are grouped as pronounced.

This is important because Hanyu Pinyin is not only a way to teach children in China to read, or a way to help foreign learners of Chinese.  It is also an international standard for rendering Chinese texts in a western alphabet.  

You should probably try to learn the actual international standard -- especially if you think you might later use Pinyin in transcribing anything for other people to read.

As to the question "why not write everything exactly as it is spoken?" Here are three answers.  But in fact Hanyu Pinyin is the outcome of nearly a century of efforts by many scholars and linguists, and I am probably missing many of their reasons.

First, so far as two or three third tones in a row are concerned, Hanyu Pinyin does tell you exactly how they are normally spoken because you know the Hanyu Pinyin rules.  What you really meant was "why not write each individual syllable exactly as spoken without relying on context?"

So, second, context is a very complex issue: As @michealyus notes, third tone sandhi on more than three successive third tones depends on how the sentence is meant to be broken into phrases.  More: the tone on a word may turn neutral at the end of a phrase, or may not if you want to emphasize that word for some reason (indeed even so common a phrase as 你好 can be spoken as ni3 hao3 in a context that calls for for special emphasis).  And the exact way a neutral tone is pronounced depends on the tone before it (see  this link).  Thus, many words would have at least three spellings.  Now suppose you as writer want to quote something someone said.  You would have to choose how to spell each sentence by deciding how the sentence would have been broken into phrases (this is often ambiguous) and how much emphasis was intended (ditto).  

Third, from the reader's viewpoint, consider the writer whose pseudonym is spoken lao2 she3.  Is the first word there a second tone word?  Or is it a third tone word being written as second?  Better to write it as Lao3 She3, and use the rule for third tone sandhi.  The ambiguities of phonetic representation of Mandarin are bad enough intrinsically.  Let's not make them worse by giving single words multiple spellings.