I stand corrected with regard to the saddles. However, I notice that all of the Elves you mention using saddles are Noldor, and the Falathrim are not. I suppose we can argue this forever; Legolas is, of course, a Sinda, like the Falthrim, but he was raised as an Avara, so his preference for bare-back may be just an Avari-thing, or it could be common to all Moriquendi.
Yes, as I posted before, both examples correspond to Noldor, and we all know the Elves of the Falas are Sindarin-Teleri, but I dont think there is anything to argue about, IMO, since the moment we have proof that "Elves riding bareback" as a general statement is not accurate and leaves plenty of room for speculation.
In cases such as this, when it comes to recreate a fantasy world with the added handicap of lacking accurate information, we are left to decide how to carry on, how to fill in the gaps (sometimes reverting to attested information as an inspiration) and aware of Tolkien's love for history, how to use this in our favour without turning ME into a fancy-looking version of Europe's Middle Ages.
I try to keep that in mind when it comes to recreate every single thing from the world of Tolkien. And so, based on the lack of information regarding Falathrim horsemanship and the gap this creates, what we know of Noldorin riders, common sense and historical background of light cavalry, I have come up with the concept you have seen.
Now, after I have explained the reasons behind this creatve decision, all that is left for me to do is hope that the concept is of any help for the development of TLA.
I am not so sure on our belief that the Sindar used exclusively axes. We know that Túrin and Beleg used swords (Túrin of course is not a Sinda or even an Elf, even in the version of the story where Beren is; nonetheless, Túrin was raised in Thingol's court and so would have Sindarin traditions, despite being mortal). Of course these were not ordinary swords. Nonetheless, I question why it was that the Sindar predominantly used axes. I can think of reasons why this might be the case, but I can also think of reasons why not.
Focusing back on the Elves of the Falas, and in some way the Elves of the Hidden Kingdom, I wouldn't say they used exclusively axes, because we have proof they also used swords. But in the case of the Falathrim, and based on historical warfare, axes (contrary to what many believe) were the most common weapon for centuries until the use of swords become more popular.
Axes are weapons related to craftsmen, not many would have a sword, but every household would have at least one axe for chopping wood, craftmanship reasons aside. It can be argued that this is history, and ME belongs to a different realm, and so it does not relate, but I believe the Falathrim to be basically craftsmen, and it goes without saying that axes would be a basic tool to their ship-making craft, making them experts on how to use axes, leaving swords as an auxiliary weapon for infantry
And perhaps a reason similar to this could be behind the use of axes by the Doriatrim, that and the fact they had dealings with the Dwarves of the Blue Mountains, and we all know their weapon of choice
Once again, this is just my opinion, and these are just concepts.
About the saddle thingie: it would seem far more logical to me, if Elves used saddles in warfare. It would give them more advantage, as saddles make for a better grip and can carry supplies, additional weapons, ammo (arrows) etcetera. Maybe even protection if they used light mail to cover their horses (I can imagine that the more important Elves did so). Of course I am generalising, but it seems a sound deduction to me. Especially Elves would care enough for their horses to be protected as much as possible in battle.
. You are right hinting at the use of mail to protect horses, but this is not always the case when it comes to historical light cavalry
However, if it is agreed the use of mail in horses, I will certainly be happy with the change too.
Btw, this was just a basic concept for a Falathrim rider, I picture their captain's horse looking different.