If you're relying on Horsemen to fight Standard units and can't get the job done otherwise (if you think that AI warmongering would be OP), I think that just goes to show how powerful Horsemen are early on in the hands ofand why they should be moved.
Not an argument. Horsemen require more production and resources than the other units so you are buying convenience with resources, which is not a bug or overpowered; furthermore, horsemen have other jobs to do, like raiding, pillaging, flanking, finishing off units, making surprise attacks from behind etc. so if someone isn't skilled enough to use regular units to keep the enemy occupied and has to rely on horsemen, thereby waging a much less effective war, then that is a problem with their skill and not with the units.
Anyway, nobody is saying all-horseman armies are common, or that other ancient units are useless it's just that Horsemen are prioritized in Ancient Era but are still dominant throughout Classical so they're just a really good overall unit. I think Ancient armies would be more balanced with Spears/Archers/balanced Chariots all playing a core role.
Your OP sure made it sound like the other units are pretty useless, but I'm glad we agree that they are not. I have never built a horseman-only army and don't think that makes sense and I don't recall seeing that in any of the Journals that have been posted here, either. If the path of rushing horsemen was so clearly superior then why are all these Deity and Immortal players not using that more often? It's because Spears and Archers already play an important role as do swordsmen and I build all of these units every time I play (King or Emperor for me, depending on what challenge level I want).
The only exception here is the Chariot Archer, which I build significantly less often, but that would be a discussion about the Chariot Archer maybe being too weak, not the horseman too strong. The reason why the Chariot Archer got lumped into this discussion is because moving the horseman back creates a hole for a decent mobile unit that then would need to be filled and naturally people looked at the CA and found it lacking.
Additionally, the fact that people are (justifiably) reluctant to accept a strength nerf of the horseman because it would become too weak in the Classical Era shows how it's not really overpowered, being a late Ancient Era unit, after all. If you want to rush horsemen, which includes choosing that specific research path over other valid possibilities as well as planting a city on horses, which of course means that you potentially give up a better position for it, then that is your choice but the vast majority of games will not be played that way (again, look at the journals), so it sure looks like you are using a bit of a straw-man here that doesn't apply to the average game flow (certainly not on Epic or Marathon).
So, again, this entire debate seems based on a very shaky foundation; the historical aspect is highly dubious, the overpowered status of horsemen seems a bit of an exaggeration, the other units are not irrelevant at all (although one can discuss the CA, you have pointed to the correct thread for discussing that already) and the idea that rushing horsemen is the best strategy in every game or even in most games doesn't match with the reality we see here. Moving back the horseman seems to create more issues than it solves (if it solves any at all) and requires extensive trialing and rebalancing. While I have no problem with working on the CA or increasing horseman production cost slightly, any more drastic changes are IMO not going to improve the game (the CA melee thing was stated by
@Gazebo to not work, unfortunately; that could have been a nice change but if it doesn't work then it doesn't work).