NO!
Like i have said to TDL. It depends how much it will cost. So if one waraxe will cost like 5 iron and 2 wood or sth like that, I think they arent that cheap,are they?
Agreed, 2 steel + 2 coal sounds fair?
Still I imagined it just as normal 1steel+1coal - please take this value as proposed in my point of view below. This way I find it easiest to implement, just making it as another equal option for weaponsmith.
I dont agree with alot of the points you mentioned above , but particularly with the one that says that they should be as weak as militia when under fire, I think they should be between axe fighter and lancer when under fire, if it has same resistance as militia against arrows, then it does not justify its cost.
I have to agree with you after analysing the situation (if we're talking about the warriors not the barbs)
The armour rating of lance carrier would be more likely for them - against arrows - against melee they'd have normal iron armour.
And I propose that for a reason that if they would cost waraxe+armour they'll be the same cost as pikeman. And, honestly, than I'd use them against both cavalry and shield infantry, why bother with pikeman.
So if their implementation will ever happen, I thing they should get their own place between existing rocks, papers and scissors. So superb melee skills in exchange for missle weakness.
And about barbarians. My opinion of them belonging to campaign is not about balance, more about aesthetics and historical setting. In single player they apear in the story. Imho they just LOOK out of place beeing recruited in medieval city. Warriors on the other hand look fine for me - still, I mentioned that if someone is looking for a nice sprite-reworking project - axes could be changed into much more likely two-handed swords.
At the end it's worth mentioning that aesthetics is probably most subiective of arguments, I just gave an opinion.
Market, although I rarelly use it, is very welcomed in a game Knights and MERCHANTS.
And nerfing missle troops with shield bonus....awww I find it most briliant change, it brought back the joy of playing after Knights and Merchants beeing Crosbowmans and Crosbowmans. Changes good for diversity, gameplay, ballance, aesthetic and realism ; )
Puny lords starving your subjects, such weakness, shame on you...
Because they are better than knights(and than swordfighers). 
