Well, the Military did shoot up one side of the conflict.
But in addition to the highly likely payoffs that occur in Mexico, it is also an economy of force measure to take on the small and wounded invading force, rather than the larger and entrenched local cartel.
If you just want to stop the War in the streets, finishing off the smaller side would work, and probably cost fewer of your own guys’ lives.
where are all the stronger-than-men women finishing off the cartel fighters?