If someone were to do it, wouldn't it defeat the whole purpose of the system by answering "yes" to this question and letting the backdoor users know that someone is now laying traps to catch them using those backdoors in the act?

However, the effort of setting this up could be better spent on making (or contributing to) a messaging app that's open-source and secure by design, where you don't need such setups to detect backdoors because they are near-impossible to introduce (open-source code and lack of a central server).