The goal is for these young people, naive and hopeful, to go out into the world and be rejected by it. The elders want these kids to repeatedly face rudeness and rejection. They want to instill the belief in the kids on mission the idea that the LDS Church alone is the moral beacon, the guiding light of the world. The only true and holy path.
The elders don't actually expect that the kids on mission are going to get a statistically significant number of converts just going door-to-door in some other part of the country. They see that the key to increasing their power, and increasing membership in the LDS Church is to retain within the religion as many kids as possible.
Sure, they'll lose some kids, but maybe those would have been disruptive anyway. If the church has done a good job of indoctrination on these kids since birth, then most of the kids will stay. And if the greater world rejects them, then the kids may think that the church is their only true home.
The cynical part that I disagree with you on is I don't think it's about "power." They really believe they have the truth from God and they care about salvation for people getting them on missions is a good way to boost their faith and dedication.
Oh, and if we're talking about the Holocaust , I find the Mormon obsession with Jews very disturbing especially after I learned they baptised holocaust victims into the Mormon church posthumously. (https://apnews.com/article/992dd887f7b948d0a08055dff0363aa4)
- A jew