One simple reason is cost. The United States already spends more money on its military than most of the rest of the world combined. The United States would benefit from having other countries shoulder some of the financial cost of keeping the world safe from weapons of mass destruction.
Another reason is that other countries have valuable intelligence information that they can supply the United States with or use themselves in order to combat weapons of mass destruction. Non-state actors are difficult to follow until they launch an attack. Other nations can use regional relationships as well as their own intelligence assets in order to make the world an overall safer place by limiting access to weapons of mass destruction.
Another reason is that the citizens of other nations may be more receptive to interventions by non-Western powers. The United States has had long-term involvements in Afghanistan and Iraq that have led to the people living in these regions having mixed feelings about the United States. If other regional powers were to act in limiting the spread of weapons of mass destruction, the United States would not be perceived as a meddling Western power. This would cause other nations to feel more favorably toward the United States.
While it is admirable that the United States takes the lead in combating weapons of mass destruction, it does not make sense for the nation to fight this battle alone. There are costs, both real and hidden, in maintaining world peace. If the United States takes on an advisory role, instead of leading in every intervention, it would be more cost-effective in terms of money and potential for combat. This would be positive for both domestic and foreign perceptions of US power.