I have been on a lifelong crusade to wipe out black widows. Bug spray, spray paint, WD40, hair spray, gasoline, fire, shoe stomping, liquid plastic, alcohol (the kind you drink and the kind that you put in a cut to remember what pain is), a box of praying mantis', vacuum cleaner, water hose. You have something new I'll try it!
They need not go extinct but if they were beaten back to live only in Antarctica as ice spiders that would suit me fine. All living things play a part in whatever ecosystem they in habit. Getting rid of snakes means a boom in the rodent population. Getting rid of spiders mean a boom in spiderless garage space, and a boom in other insects though I don't think insects are in any danger of not booming in population. Spiders are creepy *****.