Yeah, Wikipedia says there's an evidence of Roman saddles from the 1st century BC.
I've searched google images for ancient Greek horseman basrelief, and it shows everyone sitting on a bare back with no saddle or even a padding..
There is a funerary sculpture showing a Greek riding a horse with a padded saddlecloth, but it's from the late 4th century BCE, so technically is Hellenistic and may represent 'Persian' influence. Xenophon, in his book on horse-training and handling, specifically mentions the Persian saddlecloth but, significantly, not Greek ones. The Roman saddle from the late Republic may also represent 'borrowing', from the Gauls. We know from linguistic evidence that the Romans borrowed almost all of their chariot and cart technology from the Gauls (surviving classical Latin words relating to chariots and carts are all loan-words from Celtic, a pretty good indicator that they also borrowed the things they refer to) so borrowing other 'advanced horse technology' is not much of a stretch.
Don't stirrups help to adsorb the shocks of fast riding? And to stand up / bend over to reach further with a weapon?
I think they should provide some substantial advantage, otherwise why have they become so ubiquitous.
Stirrups arde a great help to a beginning rider who plans to stay firmly in the saddle, and a great 'brace' for striking downward against infantry, so they are by no means without usefulness. On the other hand, for the rider who plans to be all over the horse, like the Roman light cavalry in the Training Exercise I mentioned, or the American Native horsemen who regularly shot from under or around the horse to avoid return fire, stirrups are actually a hinderance. Different techniques. Once the lance became the Primary Mounted Weapon in Europe, there wasn't much point in dodging around on a horse, so sitting upright and braced was the most important task. Also, in Dark Age/proto-knight Europe anyone who could afford a horse could also afford armor, and wearing armor and dodging about on a horse generally means you are going to fall off a lot, or pull your own horse over with all the extra weight on his side. Either result leaves you most likely with broken bones - and an opponent falling off his horse laughing at you.
In the game, the resource system could be more dynamic, allowing to introduce existing resources to new locations. With district-like mechanics, e.g. Rice should have 2 nearby river tiles, Horses -- 3 nearby unimproved flatland plains (or 2 grasslands -- with the Shodding technology), etc
To train advanced cavalry units maybe the Stables building should be a prerequisite. Btw i wonder why there are no building-prereqs in the game
This is how Threads go rambling off in different directions, in that I thoroughly agree: in addition to revamping the Eureka/Inspiration system, Resources in general could stand some major changes:
1. Not all resources of the same type should all magically appear at once: the game designers apparently never heard of Gold Rushes, Silver Strikes, or opening new Oil Fields, but the game would be better for them.
2. Spreading many resources should be perfectly possible, with varying degrees of effort: potato, maize, horses, cattle, cotton, tea, coffee, tobacco, cacao, silk, wine - the list of 'resources' that humans have historically spread all over the globe is long and, again, totally unrepresented in the game. As you mentioned, the Requirements to spread a resource can be made more stringent than the requirement for it to originally appear on the map, or require some 'extra' Improvement, Technology, District or Building to spread and utilize a resource.
3. Most Resources have substitutes. It is much more common to haul chariots with horses, but it can also be done with Mules, and slower wagons and carts have been hauled by every animal from horses to water buffalo. A heavy War Horse might be required to mount a fully-armored Knight, but lighter cavalry can ride just about anything on four legs that neighs - in contrast to Civ VI, which requires 'horse resource' for the light cavalry, but not for Knights or Heavy Chariots!
4. IF you allow 'Resource spread' you can also introduce new mechanics for other things into the game: If every grass/plains tile can potentially become a Grain Tile for Maximum Food Production, then you can also have extra Amenities for Balanced Diet, giving the (smart) player a reason to have a mix of tiles with Grain, Cattle, Sheep, Olives, Wine and Importing Bananas (relabeled 'Tropical Fruit'?) - with more Amenity value for greater variety. This opens up the Trade possibilities as well - why shouldn't a late-game Civ make some major Gold by exporting Wine, Beef (cattle) or Grain the way France, Argentina and the USA have done in the past century?