The idea is that given an infinite truly random output of text by the nature of infinity the text of Shakespeare will be outputted in its entirety eventually

Only for a certain kind of randomness. For example, it’s possible to construct a random process that at each step emits a uniformly distributed character, but which also includes a filter that blocks the emission of the string “Falstaff” if it occurs. Such a process cannot ever produce the complete works of Shakespeare, since the complete works include that string, though it will still contain (for example) every lost work of Aristotle, as well as an infinite number of false and corrupted versions of those works.

But yeah, an unconstrained uniform-random-distributed countably infinite sequence of printable English characters and whitespace cannot be proven to not contain the complete works of Shakespeare, or any other finite sequence. I believe it’s also impossible to exclude any countably infinite sequence, but I might be wrong on that part, since my mathematics education happened a very long time ago.