I have answered some of your questions in this sub before, one of which is linked in the FAQ, and I will try my best to address the rest:

I know languages evolve naturally, as does speech. I was once told by a prof that due to the Great Vowel Shift (1350-1600’s) in England, the way we speak in (parts) of North America is likely closer to how people spoke Shakespeare’s time than how the Brits speak today.

This is a myth debunked at length in Language Myths ed. by Laurie Bauer and Peter Trudgill, in the chapter "Myth 9: In the Appalachians, They Speak Like Shakespeare" by Michael Montgomery, a professor of English and Linguistics at the University of South Carolina. You can read my previous response in this sub to this myth here. Montgomery's article even goes into the origins of the myth. My previous answer also quotes from another book, Word Myths: Debunking Linguistic Urban Legends by David Wilton, and I'll quote it here again for convenience:

"All dialects change over time. Most will have some relics of Elizabethan language that have fallen out of use elsewhere. Those that are isolated, like Appalachia, may retain a few more archaisms than dialects that have a lot of contact with the outside world, but even these isolated dialects change. The mountain speech of Appalachia or the Ozarks is no more like Elizabethan English than any other dialect, even if a few words or the occasional grammatical structure are similar.

"Still the lure of this legend is strong. Those who speak non-standard dialect are often stigmatized. They are viewed by outsiders as rustic and uneducated. It is no surprise that they are attracted to a tale that connects them to a great literary tradition."

Onto your second question:

So when did American English evolve from British English,

I have addressed this question here, which is the one linked in the FAQ, and with some further information here.

Briefly: American English began evolving just as soon as there were English speakers living and being born in America. By the mid-1600s, British/English travelers had already commented on a handful of differences in word usage of the Americans. By the last quarter of the 1600s, there is circumstantial evidence that the British/English were already recognizing wholly unique American manners of speech. By the end of the 1710s, there was the first direct mention of an American manner of speech in two books published in 1724 written by Hugh Jones, who had been a visiting professor to the College of William and Mary from 1715-21. In 1739, Benjamin Franklin wrote in Poor Richard's Almanack about the difference in dialects between various American colonies. There had been other mentions as well, so various American manners of speech were well-established by then. By the time of the American Revolution, American and British writers were commenting regularly on the differences between them in their manner of speech. More information is provided in those previous answers.

and more curiously, why?

Sociolinguist Peter Trudgill answers this question directly in Chapter 1 of New-Dialect Formation: The Inevitability of Colonial Englishes, which you can read in full here. To summarize the six points he makes:

1. Americans encountered new plants, animals, geography, and topography not encountered back in Europe necessitating the adoption of new words.

2. After colonization of America began, the English back in England continued developing linguistic changes, as languages do, not all of which were imported to America.

3. After colonization of America began, the English speakers in America began developing their own linguistic changes, as languages do, not all of which made their way back to the British Isles.

4. In America, English speakers had contact with indigenous people which contributed to the English language and vocabulary there.

5. In America, English speakers had contact with speakers of other European languages which contributed to the English language and vocabulary there.

I'll quote #6 verbatim, since Trudgill points out its importance and probably most directly answers the question:

[6]. From the very beginning, American English was subject to processes associated with dialect contact. Although of course the geographical and social origins of the settlers were different in each location, none of the early anglophone settlements on the east coast of what is now the United States was settled from a single location in England. We can therefore assume that, very early on, contact between different British dialects would have occurred in the American settlements, which would have led to the appearance of new, mixed dialects not precisely like any dialect spoken in the homeland. These were all, as Kurath has said, ‘a unique blend of British types of speech’ (1949: 1). The fact of modern regional variation along the east coast of the USA is thus explained not only in terms of different linguistic changes having taken place in different areas during the last 400 years, but also more crucially by the fact that the initial mixtures – and, therefore, the outcomes of these mixtures – were different in the different places from the very beginning (see Algeo, 2001). In the words of McDavid (1985: 19), we can assume ‘dialect mixture from the beginning in each colony...with different results’.

For the rest of your questions...

As a Canadian (British English) I’ve always found the more phonetic American English very strange.

Centre vs Center Phase vs Faze ...

This question is addressed in the article/chapter "Spelling" by Richard L. Venezky that appears in The Cambridge History of the English Language Vol. VI: English In North America ed. by John Algeo, and is also touched on in other chapters in the book. When English-speaking America was first settled, there really weren't any comprehensive English spelling books, let alone dictionaries. Initially, American spelling was the same as British spelling, as best as they could figure it, and "[f]rom these beginnings, the solidly British core was occasionally expanded and less frequently replaced to yield the orthography that prevails today". In both America and England in the 1600s and early 1700s, spelling was much less stable, with many words having multiple different preferred spellings. It wasn't until the early to mid-1700s that English spelling began to become more uniform, with the widespread publication of dictionaries. One of the most important 18th Century dictionaries was A Dictionary of the English Language by Samuel Johnson, published in London in 1755 and reprinted (and often pirated) in America.

Consequently, up until the Revolutionary period, British and American spelling was mostly the same among fully literate people. However, as Venezky writes, there were a handful of American preferences even before the Revolution that weren't used back in England. Further, before the Revolution, in both Britain and America, there began a movement toward English spelling reform, such as leaving off the ending -k in words like logic, music, and physic.

During and after the Revolution, there arose a more deliberate spelling reform movement in America to "simplify" the spelling of particular words. This movement was led by lexicographer Noah Webster and "[i]t was...Webster’s 1828 dictionary, An American Dictionary of the English Language, that established the majority of American spellings now distinguishing American from British spelling":

Final -re was replaced by -er (scepter, theater, meter), ou by o (honor, favor), -ence by -ense (defense, pretense, recompense), -ize was generalized over -ise (methodize, patronize but criticise, circumcise), and final consonants in words like travel, cancel, and worship were not doubled before suffixes beginning with vowels. In addition, a number of spelling preferences, such as wagon over waggon, distinguished Webster’s American spellings from prevailing British norms.

Webster also introduced spelling reforms that did not catch on and were reverted back in subsequent editions of his dictionary, such as "ake, aker, bild, bridegoom, nightmar, tung, turnep, wo, iland".

(To your question: faze and phase is new to me if that's how it is in Canadian or British English--I thought those were two separate words, as they are in American English. If true, I'm not sure when that one was introduced.)

Was it a big “F You” to England?

That would be overstating Webster's intentions. In the article "British and American, Continuity and Divergence", author John Hurt Fisher writes, "By 1787...Webster had been drawn into the nationalistic movement...But it is hard to know the extent to which his spelling reform was intended to distinguish American English, and the extent to which it was merely a play in the international game of rationalizing English spelling."

Was it a school system issue?

No, not really, except for the fact that Webster's intentions were for his dictionary to be used nationwide across the United States, which presumably would mean it would be used in the classroom.

Did it have to do with teaching a very complicated language to natives?

No.