You are arguing from the typical, venerable, valid-in-post-enlightenment-culture dynamic in which ideas propogate on their own merit. In such an environment, the adaptive strategy for dealing with bad ideas is to ignore them and let them die on their own lack of merit, or "burn out" as you put it. This has been known to work as long as the immunity mechanisms against bad ideas are intact, and are capable of dealing with the particular strain of ideological pathogen.
This strategy is not effective against ideologies not in that set, such as ones that spread themselves by the sword or by intimidation or in populations that believe ideas for reasons disconnected from the merit of the idea itself. The fact that said ideologies are bad or invalid or incompatible with civilization as we know it is not directly relevant to whether they will catch on or not. Such ideologies have spread themselves very effectively and they don't care that they are regression or that they are harmful. Effective disease agents dont care that they make the host organism sick, they are effective by definition if they propogate themselves, and just ignoring the symptoms and counting on typical immune mechanisms to make them go away doesn't always work.
Westerners think that ignoring ideas or debunking them is going to always work, but those techniques only work in certain contexts and against certain threats. Ghandi's techniques were only effective because he was operating against the British Empire and pushing their buttons, for example.
The West is not able to fight back bad ideas because, sometimes in an attempt to stop low level autoimmune problems, she has ingested massive doses of immune-suppression; immune mechanisms such as the nuclear family, shared but diverse Christian heritage, societal structures are weakened, made obsolete by technology, or dismantled, and ideological infections thought to be conquered are breaking back out in the unprepared populace, and getting some rest and drinking some fluids until it burns out may not work.