Jun Moxie wouldn't have agreed irrespective of what his uncle would've said—except that he mentioned that incident. And this moved the Young Master Jun deeply.
Jun Moxie's heart shook. He suddenly remembered the manner in which Jun Zhantian had issued a call to war; the manner in which his grandfather had led his troops to commit that purge in the capital. This memory suddenly warmed his heart.
This memory moved Jun Moxie the most.
Hit-man Jun was aware that his third uncle's decision was based more on sentiment than reason. However, it would result in a lot of harm to the Jun Family if Jun Wuyi truly undertook that task—regardless of his success in saving the Princess. However, the heavens find a way when a man of noble character sets out on a righteous task.
It would be difficult to cover-up his uncle's involvement if the well-known general was to get involved in the matter. However, if Jun Moxie himself undertook the task, then…
The title: to get someone else to do one’s dirty work/ to kill with a borrowed knife, is an art of Warfare Deceptions. More details can be found in the 36 stratagems of Chinese warfare.