EDIT: Just a runaway thought: why is biodiversity important? If we are optimizing for the general well-being of humans and living beings on earth (which is the frame of reference I am using), then extinction of a species while increasing experiential diversity seems worthwhile.
In addition, the extinction of a species is only sad to those who remember. Many cultures have merged in history to become nations we see today, but nobody misses the oh-so-glory days of a specific Northern German tribe in BC 500.
As to the beauty we experience from diversity of species (seeing them, interacting with them), I would much rather enjoy technological experiences over that natural experience.
Perhaps this makes it sound all too all-or-nothing-esque, but it seems that there is a general assumption that increased biodiversity is good beyond measure.