In one of the books (I think Prisoner of Azkaban) the Ministry of Magic puts out a bulletin to stay away from a fugitive after informing the MoM and to let a Hitwizard team take care of them. When you've got dark magic and shit having dedicated kill teams makes a lot of sense, TBH. Aurors are, IIRC, magical detectives, equivalent to the upper-ranks of the British cops with a lot of lower-level guys acting as basic plods, as we see from Arthur Weasley's constant issues with his unloicensed everything he only barely dodges thanks to his connections as a government employee.
As to other duties that don't involve other wizards, there's monster hunters because of course things need to be dealt with, non-lethally or permanently, and the non-shitty adult Weasley child is a curse breaker for Gringotts who raids ancient tombs and shit for their loot, D&D style.