First off, we should acknowledge that your argument has shifted significantly - from an ontological to a practical concern. The ontological question, whether the archive exists or not, is independent from the practical question of whether or not it is useful. Initially you said that no archive could contain more than ~10^80 objects because the number of atoms in the universe placed an upper limit on its contents. Now you say that this website does not contain its books because of the amount of work necessary to retrieve them. Would you consider this to be different if the quantity of texts was less than 10^80? If not, we should recognize that you have shifted from your earlier argument, which was untenable.
As for your present argument, i would first point out that you should not treat quantitative differences as essential differences. Any archival system requires some sort of key system for retrieval. Print libraries have the Dewey decimal system, the internet has URLs, and libraryofbabel.info has its book locations, strings of letters and numbers which you are correct to point out are just as information-rich as the texts themselves. But you claim that because of the size of keys, the texts in this digital archive do not exist - this is fallacious. How large must a key-value system get before it ceases to exist? If a poem's title is longer than its text does it cease to exist? Greek philosophers parodied the confusion of quantitative and qualitative thought by asking how many grains of sand we needed to add together before we had a heap. Once again, you have confused a practical and an ontological argument - the length of keys (and quantity of text) may affect the usefulness of libraryofbabel.info, but they cannot affect its existence.
You consider three heterogeneous things in your examples - a floating point register and the set of possible floating point numbers, libraryofbabel.info and the texts it contains, the human mind and what it "actually thinks about". Let us begin with the last - you claim that the human mind only contains what it "actually thinks about." If this is so, what is a memory? Your failure to grapple with the difficult category of possibility leads to the ontological confusion in your argument.
If we limit our consideration to practicality, I would agree that the libraryofbabel.info is not useful in the same way as a digital archive like JSTOR. In other libraries, one can find texts by subject matter, author, period, etc. and learn something new from their contents. libraryofbabel.info exists to show us the same language we have encountered already in a new context. This allows us to reflect on the essence of language, which is not restricted to the intentions of a conscious and rational speaking subject (as you imagine when you speak of what we "actually think about"), but is always open on irrational excess.