a CR3 axeman will take down a defender that might cost 3+ regular axeman. Look at it this way:
CG1 archer in a flatland city with 20% cultural defense, totally fortified has a strength of 6.45 (plus its first strike)
CR3 axeman has a strength of 8.75, and you have a very good chance of taking out the top defender in one go. If you attack with an unpromoted axe, your first one dies almost guaranteed, and then you have to send another unpromoted axe to die against the next full health archer. Then there's always the scenario where you attack and die but do little or even no damage, and if you dont take out all the defenders in one turn you are facing a healed and promoted defender the next turn.
And yes, the barracks only gives you the first promotion. But the first promotion makes getting to level 3 so much faster and easier.
Also remember that if you decide to settle your first great general, which you tend to get pretty fast, you will be pumping out axes or swords that start at CR2. Not sure how many people actually settle though, it seems that super medics are what most people like to do.