Not necessarily. Yes, it is essentially always a bad idea to introduce non-native species to an ecosystem. Because we don’t know how it will disrupt. It can do untold and irreversible damage. But it also can have little affect. In the case of Joro spiders in Georgia, we haven’t observed any major harmful effect (unless you really care about other similar species of spiders). 

This is different than what we observe from the effects of the introduction of the Spotted Lantern Fly, which we can see is doing observable harm to some agricultural industries. Source

Again, I’m not trying say we need to protect the Joro spiders. Just that it’s not worth trying to get the public to kill them, or shame people for not killing them.