I'm...not sure why exactly Ajax liking to fight and not being one to back out from one or getting rid of dangerous monsters for people who ask him to is a problem when there's characters described as assassins and the like and Ajax is specifically said to be an okay guy to be around with if you don't try to fight him.

Most traditional heroes made their impact precisely because they presented these attributes of foolhardy courage and relentless battling both for self improvement and against threats towards either humanity as a whole or their lands/close ones. It's not the killing that matters, it's the test of personal strength and skill, with no tools or anything to facilitate success, that matters most.