I tried to look for solid evidence that the story was fabricated, but could not find any (nor could I find anything to prove it is true). A few articles (e.g., https://www.mountvernon.org/library/digitalhistory/digital-encyclopedia/article/cherry-tree-myth/) nicely summarize how the story spreads around, but they simply claim that the original story is true or false. If cherry tree is impossible, could it be another plant easy for a 6yo with a hatchet?
Personally, I have always liked Grant Wood's version, with its echoes of the famous self-portrait of Charles Wilson Peale.
Although major events are often well-documented ( like Washington accepting the surrender of the British at Yorktown) absolutely proving trivial events did or did not happen can be quite difficult because typically no one bothers to note them at the time, and/or there are few or no witnesses. But we can pour a lot of doubt on to this. Here, there are two possible sources, Washington and his father Augustine. . While I don't know what records survive of Augustine ( he died when George was only 11) a lot of George's documents survive, including school exercises. There's nothing surviving from either of them that verifies this story. The story only appears after Washington's death, and only in one source - Parson Weems' bio of Washington. And, as the Mt Vernon site notes, only in a later edition, that of 1806. Weems' book was also regarded as dubious even at the time it was released: another page on the Mt Vernon site quotes one reviewer as calling it "eighty pages of as entertaining and edifying matter as can be found in the annals of fanaticism and absurdity". The cherry-tree myth was not the only one that Weems told, and of course Weems provided no names of sources for his information.
If George Washington cut down a cherry tree when he was 6 and no one wrote it down at the time, do we have to consider it might have happened? Sure. But we can also say that because the guy who first told the story had no sources for it and had good reasons to make it up, we don't have to value it as anything more than a story. Much like the lives of saints written hundreds of years after the saints had died by monks who had few or no records to consult, the purpose was not to reveal the truth but to inspire the audience.
As to whether a 6 yr boy old could chop down a cherry tree with a hatchet: we are not supposed to quote ourselves as authorities, but a 2 yr old Montmorency cherry sapling is going to be perhaps about as thick as your thumb, and some rifleman's hatchets were actually rather small and would have been easily handled by a child.
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EDIT In checking Weems's cherry tree anecdote, I find " an aged lady, a distant relative, who when a little girl spent much of her time with the family" was the source of the previous anecdote, also about fruit trees, and so perhaps Weems implied she was the source for this one also. As the dialog between Augustine and George is quoted at length, the "little girl" must have been old enough to write. Judging from the length of Augustine's moral lectures fully quoted by Weems, she would have had to know shorthand as well.