Society destroys its mockingbirds in a few ways in the book.
First, society destroys it mockingbirds by having blindspots. The power of a blindspot is that you do not know you have it. When a society has a blindspot, it can be very detrimental. When we take this insight and apply it to Maycomb, we see that racism is a blindspot. The very air Maycomb breathes is filled with racism. Even the missionary society does not know that their views of race belie what the Bible says. Here is another example. The Cunninghams from Sarum would have killed Atticus, even though Atticus helped them. On account of things like this, Tom Robinson was destroyed.
Second, not too many people have true courage. One underlying point of the novel is the true nature of courage. True courage is to stand up for what it right even if you will lose. Atticus demonstrates it, but the problem is that there are not enough people like Atticus. The outcome once again is that mockingbirds perish.