How different would politics be if every representative had to spend a few months in the military followed by a few months in poverty?
In the post-World War II period, most successful politicians served in war. Some came from poverty (Lyndon Johnson and Bill Clinton come to mind, but so did Abraham Lincoln).
There is no evidence that those who had seen war were less inclined to send others to war. As for poverty, those who experienced it were more inclined to promote efforts to help poor people; yet no one produced as vast a set of social programs as the very wealthy Franklin Roosevelt.
In sum, I am not sure that there is a hard an fast cause-and-effect dynamic between experiencing war and poverty and leading in a way that avoids or diminishes either.
However, it seems more honorable for a person commanding others to risk life and limb to have done so, just as it seems more equitable that someone who has struggled with need should weigh efforts to alleviate poverty. For that reason alone, it might be a worthwhile requirement in a democracy.