For MCH, don't you prove yourself to Ishgard and the one noble house trying to stop you throughout the quests?

There are certain stereotypes to each job, sure. But one thing to remember when it comes to RP is that your character is an individual, not just a homogenized collection of stereotypes and cultural norms.

I would imagine that yeah, warriors in general probably rely on raw strength and rage while a paladin's techniques are probably in general more refined...but that doesn't necessarily mean that pertains to every single individual. So you don't necessarily have to stick to it in your own headcanon.