I should have said "at least until" rather than "until", sorry. It's from a result that predates the paper, due to Fisher (using butterflies, not code bugs).
The actual guarantee from the result is not that the number of unobserved "species" is small, but that the total population of all unobserved species is small. If you go back to the birds example, then you could say something like "at most 0.1% of all birds are from species that we haven't identified" but maybe those 0.1% of birds are from a million different species each with incredibly tiny populations. In the code bug example, the very rare species would be the bugs that are very unlikely to be found, i.e. it's more about estimating how many more bugs you will find if you continue to analyze it than how many are really there.
The actual guarantee from the result is not that the number of unobserved "species" is small, but that the total population of all unobserved species is small. If you go back to the birds example, then you could say something like "at most 0.1% of all birds are from species that we haven't identified" but maybe those 0.1% of birds are from a million different species each with incredibly tiny populations. In the code bug example, the very rare species would be the bugs that are very unlikely to be found, i.e. it's more about estimating how many more bugs you will find if you continue to analyze it than how many are really there.