I doubt the kid was dumb and Fermi never tried to prove anything. This was a conjecture he made in a conversation with other mathematicians. Some thoughts on your post:
Without knowing how many dead languages and when not counting languages with higher amounts, I got 1.199e+20 (or 1.199 x 10^20). I am told that the number of stars is something like 10^24, so I am pretty sure if were did the real number (not an average) of words in each language, and added in all languages, this simply would not be true.
Just taking your math for granted, you have proved that it is true. The difference between 10^20 and 10^24 is huge. Even if you were off by a factor of 100 due to your assumptions, it would be huge.
The thing is that if Fermi's idea is that there are that many stars, how would this "disprove" aliens? If anything, this establishes a number of possible worlds for them to live in if there are more stars than languages and people. But even if there weren't, in a single world, we have millions of species of creatures, not just creatures themselves but species.
Ok, where are they? That is what Fermi asked. So many worlds, so many potential species, so much time (our solar system is not particularly young or old), so why no aliens?
Fermi did not say "no aliens". He hypothesized that intelligent life is actually very rare or very fleeting and combined with the distances involved in the universe contact would be rare. He came up with that because if intelligent life was very common, we should see it. And if it was very persistent (cultures did not destroy themselves), we should see it since there are systems millions of years older than ours.