I'm not sure if I think it's the narrative that's biased against Paris - maybe more the commentators? e.g. wikipedia says "Homer's Iliad casts Paris as unskilled and cowardly."

He is certainly portrayed as not a warrior by nature. Paris doesn't much like war. The anti-Paris, Menelaus, by contrast is repeatedly called "war-loving" - but I don't think it's intended as a positive attribute.

I don't think the narrative likes war, and doesn't like those who push for war (as opposed to those who honorably respond when called to defend - i.e. Hector). Book 4 characterises war as "hateful war" - when the gods stir up the reluctant armies to fight:
"With them came Terror, Fear, and tireless Strife,
sister and companion of man-destroying Ares—
at first small in stature, she later grows enormous,
head reaching heaven, as she strides across the earth.
Strife went through crowds of soldiers, casting hatred
on both sides equally, multiplying human miseries."

And though it's brilliant about the excitement of seeing great battle-hosts move and clash, still the most powerful, lasting, mood of the poem (isn't it?) is grief for loss - first of Patroclus, then, devastatingly, of Hector.

(I don't like Achilles either. I do like Hector, though, and Patroclus.)