The short answer is that promotions don't work as simply as they first appear. All else being equal an Axe will have better odds to win against a Vulture, and a Dog Soldier will
destroy a Vulture.
EDIT: To clarify as best I can, Combat promotions simply add to a unit's own strength (so a Combat 1 Vulture has 6×1.1=6.6

, Combat 2 has 6×1.2=7.2

, etc.), and other modifiers are applied to the defending unit's

(either positive or negative depending on whether the final modifier is positive or negative).
In the example of Vulture vs Axeman, with no promotions or terrain in play, an Axe attacking a Vulture gives the vulture a -25% modifier (-50% from the axe, +25% from the vulture), meaning 5

vs 6/1.25=4.8

, giving the axe ~65% odds to win. A vulture attacking an axe will give the axe a +25% modifier (-25% from the vulture, +50% from the axe), meaning 6

vs 5×1.25=6.25

, giving the vulture a ~35% chance to win.
Because of this interaction there are situations where taking Combat promotions can actually make more of a difference than taking the relevant -25% vs. specific unit type promotions, or City Raider promotions. A full health Praetorian gains a flat 0.8

per Combat promotion, whereas taking Cover/City Raider II only takes away +25% from a defending Archer, which translates to 0.75

. If you're using Catapults (or other siege units) to damage defending stacks you can likewise be better off with either straight Combat or First Strike promotions in some cases, since even the +100% vs. Mounted units that a defending Spear/Pike gets is only meaningful so long as it has a decent

remaining to benefit from the high multiplier.