The biggest suggestion I have here is don't worry about having mechanics to represent everything. You don't need mechanics to say that they have a special bond with their mounts. But if you really must, just say that they're especially well trained and give the rider advantage on all animal handling and loyalty checks. You could even make a special Talenta background available to halflings only that mimics the outlander, but the special ability is advantage on specific rolls dealing with their mounts and others of their culture.
If you really want dragonmarks like they were in 3E, just import them directly and correct the pieces that don't work. 5e characters don't get a feat at first level, but that's easy enough to say that every PC get's a free feat at lvl 1 for your game and poof, just like 3E Eberron if I recall correctly.
As I said before, I don't necessarily sweat mechanics. I am fine with quick and dirty imports, backgrounds, and whatever. But some things, such as the Artificer, are VERY difficult to implement with current 5e mechanics. Even the most recent UA version doesn't really represent the Artificer's unique ability to imbue spells into objects or improve weapons. For me, this is part of the hobby of D&D to homebrew and make adjustments and build new stuff.
Class =/= Culture. I can't emphasize this enough. The barbarian class is not the barbarian culture. The outlander background best represents the barbarian culture and backgrounds do a lot more to shape a characters outlook on the world than class does. I can make an acolyte barbarian and he's a religious zealot who flys into a rage, or a mage apprentice (don't remember the name of that background) barbarian who quaffs special potions to instill his rage, etc. Having an outlander fighter, ranger, rogue, barbarian, cleric, whatever is so much more appropriate for the barbarian culture than to pigeon hole everyone who lives off the land as the barbarian class.
I never said class equals culture. In 3E Eberron, most people don't even have class levels outside commoner or adept, so you don't even need to make the assumption that Talenta Barbarians are mechanically barbarians. And as I said before based on what's presented in the Races of Eberron book, most Talenta Halflings with class levels are rangers or rogues. But also, as I mentioned previously, in the ECS the described leader of the Talenta Halflings is statted out as a 9th level barbarian. It's right there in the entry on the Talenta Plains. This leader was known for his prowess as a battle rider. If you wanted to play a character like him, how would you go about it? Sure, you can just give him the outlander background, maybe just give him a generic dinosaur as a mount, or make him a beast master ranger.
But the same question arises when someone wants a warmage. They could go evoker wizard, or Eldritch knight fighter, Bladesinger wizard, Pact of the Blade Warlock, or even Valor bard. There are multiple ways for mechanics to work for general concepts. So why make it seem like creating new mechanics to represent concepts is so bad? Isn't that the whole point of the recent UAs? Of course you can always use what your given, but D&D is inherently modular and meant to be built around one's table and play style.
If you really want to make homebrew mechanics for everything, I wish you the best of luck. I'd rather just wing it to get the feel of the setting than fret over mechanics and how/if they are balanced or fair.
I've already mentioned this is my preference and how I use the game. Good for you for being satisfied with what's given to you. But I was answering the question asked on the thread for what I would prefer. Disagree with me, fine. But you don't need to critique my opinions as if there's a right or wrong answer.