It's hard to hit with HAs before the enemy has metal. Sweet when it works, but not necessary (dunno about Deity). Difficulty seems to depend more on who the enemy is---eg Ramses with metal is still easy as pie, but Shaka with metal is tougher. But from my experience up to Immortal, HA rush into metal units is still very doable.
The only reason I wouldn't go for HA here is you have a churchill with protective on the one hand and an Augustus with praets on the other, both of whom are kinda far away. Going for one would cede land to the other, land which could be yours peacefully. I'm not an expert, but I'm looking at this thinking, if not cuirs, then rifle bust with your capital as a killer globe theatre spot. Either lib toward rifles and, assuming you're still facing longbows, attack with rifle treb (upgrade from treb to cannon costs 80g, super cheap as far as upgrades go), or lib your way to steel with all the scientists and go kill somebody with musket cannon while teching to rifles. It'll be interesting to see what the pros recommend