I have read it, without preconceptions. It is very confused. Maybe you should read Cantor's proof without your preconceptions.
From the pdf: "He [Cantor] concludes E0 differs from D for all v in the set M"
No, he concludes that E0 differs from Eu for all u in the list that E0 was derived from. Neither "v" nor "u" are in the set M, they are natural numbers not binary sequences (rows). What Hudson calls "D" is the diagonal that is negated to get Cantor's E0. Both are in M, by the definition of M. And nothing described as "all <whatever> in M" are referenced in this part of the proof.
The rest of the paper refers to constructions that have nothing to do with this conclusion, so they are irrelevant. Whether or not "A different random list could contain E0" is irrelevant, since any list you can produce can be used to construct a different E0 that is not in that list.
And the point is really quite simple: The actual diagonal argument does not prove that M is uncountable, it proves:
IF you have a list of sequences from M, THEN we can construct a sequence that is in M, but not in this list.
In "analyzing" this proposition, there is no reason to refer to "all M," or any other sequences besides 00 derived from the list in hand. But once you have the truth of this proposition, it "follows immediately" that M itself is uncountable.