Most of our handwaving about species extinctions is a tempest in a teacup if one is talking at the ultimate levels of ecosystem collapse and thermodynamics of energy flow.
We could probably remove all sentient life from the land surface of the world and basic energy flows through plant life would go along lightly altered. There would be some kudzu-like releases in plant behavior, some highly specialized pollination needs would not be met and some species would fade away, but others would fill most of their sun gaps. Most herbivory is insect work and most decomposition and nutrient cycling is microbial. Just don't mess with rotifers, diatoms, or krill. Bad system collapses would start happening.
This doesn't mean that I don't care if vertebrate species go extinct. What it means is that the real motive behind our chicken littleisms is being driven by human values and aesthetics. Some of the farmers can make an economic argument, and some hardship will be wrought on some herder/hunter groups but for the other 99.9999 % of the world's population, they won't miss a meal.
Maybe it is an inherent or even instinctual conservatism (brrrr. . . .) that helps us hate change and loss of things natural but it is rarely a survival issue.
If we are to grieve, let it be for the actual, physical loss of land bases - those eroded into the Louisiana bay bottoms, entombed under powdered mollusc shells and rebar, those buried by 100 foot shifting dunes of sand brought to the surface by plows upwind, those under asphalt, and those that are wiped to bare, exposed dirt every year by Steiger, Massey Ferguson, and Deere.
We can create an intellectual crisis and point to our [highly symbolic] grizzly bear posters, swift fox screen savers, and whooping crane calendars, but the loss we most fear is the loss of appreciation of experiencing these things and the knowledge that we share the blame for their non-existence.