But HA are almost twice as expensive as axes anyway, so what does it matter if you lose 4 axes instead of 2 HA? Also, CR axes ought to have better odds than flanking HAs, which also means doing more damage. (combat odds, not survival odds)
Axes have 1 move. Horse archers have two. As they ignore first strikes, HA's have at least as much chance if not more to deal significant damage on the archers (archers are quite poor vs HAs, you can just use combat II guys and even if the city has 20% D they'll still be around 75% win chances).
But back to that 1 move vs 2 move thing. Discounting that is painfully ignoring the greatest advantage of the mounted line. I doubt most people understand what it means when they plan a war.
2 move units literally mean you fight less troops to begin with. 1-2 less per city whipped if you don't use siege with your 1 move guys, but 3-4 less if you do (though siege can then help). If you're attacking (and losing) axes to archers, it's going to add up vs the significant increase in archer #'s. If you are looking at this across conquering four cities or so, the HAs start looking much better - leaps and bounds better than the axes unless you involve siege, but even if you do the outcome of which is better isn't entirely clear and depends on the opposition. Oh and...HAs come way earlier than siege if you prioritize them. The tech even trades decently.
Comparing the two is a little iffy though. The two are at their best in different windows of time. You'd axe rush if you are extremely boxed in but have copper. If you have horses and room for more cities/etc, HAs are definitely superior units. You said long ago that draw goes to the defender, but that's only true if the defender isn't ultimately killed on that turn...otherwise withdraw favors offense! Which unit survives and gets XP? Not the dead archer.
Basically, 4 axes =/= 2 HAs in terms of survival chance. The HA advantage is magnified...and quite greatly...by AI whipping tendencies and the way it spams units out of its cities after war...but only if it still has them.
Oh, HAs are the same hammers as cats too, and tend to survive more frequently. If the city has <4 defenders, and one is pretty tough...well i'll let you do the math.
I can't understand anyone axe rushing on Emperor or Immortal. It just doesn't work for me either - sure I can cripple an AI and take another capital - but I actually get less land than if I had REXed normally and my position relative to the other AIs is poorer than if I had done my normal expansion.
Some maps are kinder than others. The more room you have to grab cities peacefully, the less appealing the axe rush. I can easily describe a few situations where I could/would pull it off on either of those difficulties:
1. Boxed in to 3 cities or less: Not much choice here...you're probably on a peninsula. Hope you have copper.
2. One AI on the continent/landmass, and it's near you: Pretty straightforward too. You kill this AI, and you get 15+ city sites to settle at your leisure with nobody taking them in the anywhere near future. Might as well get that AI out of the way. Maybe it's different on deity, but IMO getting 15+ cities on immortal is game-breaking. A game breaking advantage that soon? I'll take it.
3. You spawn next to a non-protective leader that is virtually guaranteed to attack you: Basically any close-spawn to a warmonger. The war mechanics work in such a way that if you'll be the only civ bordering with one of the monty/shaka/gk/alex type asp-hats, you're going to be attacked. You're the first target they will look at every time they roll their RNG declare-odds. They're likely to roll an attack before you get them to pleased, much less friendly. Note that if they border someone else, getting them to pleased early will make them almost certain to hit someone else. But if it's just you, it's probably going to save hammers to just kill them. It's not like you can settle 0 cities after the axe prep. Sure, you'll have a bigger AI to deal with later somewhat, but hopefully it isn't one of the other PS loving idiots who will also declare (if it is, don't rush if they border you unless desperate, or you're a sitting duck and they will see you as one).
I find I axe rush fairly infrequently now, but I've seen maps that warrant it.
Of course, if you play on marathon, you'll probably want to take advantage of the free heroic epic in every city as soon as possible. The AI buildunitprob is set for each AI regardless of speed - the human's is not. A free HE in all cities means that much of the AI spam advantage can be erased by raising one's own tendency to build units...the AIs just don't take advantage the same way.