Similarly: if you write a program to randomly run through all the combinations of pixels on a decently large screen (say, 1080p) you will eventually see every important question and answer that can be expressed on a screen.

The short answer is no, and even if we could, the digit index you'd start at would have a larger binary representation than the actual data you were trying to encode.

Could we already do this by leveraging the Library of Babel?

Genuinely asking, I'm not really sure.