“I believe I heard somewhere that if one is davening for others with intention of getting answered first, that it doesn’t work.”
People don’t understand why it should work, so they say it doesn’t work. Nevertheless, things in the spiritual realm do not necessarily have to be understood by us mortals.
It works because it says it works. It does not say anywhere it has to be lishma. (except by people who think their logic determines if it works.)