Neanderthal
Arcane
The problem is that doing any of those things would be pretty fucking dull. Hunt for a fabled indian treasure? How? Digging up random countryside or graveyards? Do I choose whether to spend points on my shoveling skill vs my shooting unarmed men in the face skill? What would hunting down a notorious criminal entail exactly? 6 hours of talking to people and 6 hours of uneventful horseback riding, culminating in a 30 second shootout?
Save-the-world quests are overused because they're one of maybe 2 or 3 viable excuses for a party of murderhobos to leave a wake of corpses knee deep for hours on end. If you're only going to spend 1% of your game in combat, you probably shouldn't even bother having a system for it. Which begs the question, what is 99% of your time going to be spent doing? You might have a profession and maybe 1 or 2 more incidental skills at best. Are you going to wring 20 hours out of pulling teeth, prospecting for gold, and playing the piano? Sounds like some boring shit to me. Is your character going to be some magical jack of all trades that can blow up bank vaults, ride away, sell the gold, buy cattle, drive them across the country, sell them, sneak into a mansion, kill some guards, steal a deed and open a dentistry practice to fulfill his lifelong dream of yanking teeth out on top of dead indians? Maybe we could just skip the bullshit and make it a visual novel titled "Gary Sue, the westernfag game". After all, it's not like you're going to have meaningful systems for any of that shit. Probably a bunch of lame QTE or some dialogue options that make no real difference.
Well me personally i'd make it somewhat like Fallout 1 or 2. Hunting for a fabled Indian treasure sounds like the basis of a damned good adventure, first you've got how to gain the rumour, from an old drunk Indian you buy a drink for perhaps, maybe a dying Indian Fighter on your operating table, or an off his head miner who is mad as a hatter. Gain the rumour in some manner anyway, then you've got to follow this up, clues in old journals, held by bizarre characters on the open range whatever. You add a touch of danger at some point, rival treasure hunters, Indian Warriors objecting to the Palefaces presence, a pack of rabid dogs. Shovels i'd make just a use item for anybody, not link to a skill. Horses i'd treat in the same way as the Fallout car, with chances of random encounters on the range to add interest. What the treasure is, what it's worth and what it does if anything is another factor to add interest.
I'd definitely have a system for combat, rifles, sidearms, knives (throwing and melee,) bow and arrows, dynamite, held even emplaced Gattling guns if you're lucky enough. Combat is a staple of the genre that you can't omit. Gameplay should be fairly much the same as a Fallout game, except probably with more of a homebase somewhere, add content and reinforce every skill choice, make interesting chocies and consequences. Give the protagonist a motive to move towards and feed him scraps of info and hints that allow him to do so.
Say as a Doctor you pull a tooth first thing in the morning, gain a few bits. Next you have to choose whether to take the leg of a man whose been shot, try and dig out the bullet or leave it in there and these all lead to different consequences. At noon you could go and have dinner at the salloon where almost any encounter can happen and you have drinking, gambling, brawling, talking to pick up info, playing the piano, banging a whore or whatever to amuse you. In the afternoon you could pull a baby out, treat a dose of the dick rot or whatever. All of these uses of the Doctor skill would require tests against your skill and have consequences, if you keep failing you could be run out of town or replaced. Or you could just shut up shop for the day and pursue personal business.
There's no need to choose such a static life though, you could be a traveller and only pursue your skills where and when you choose to, there should be plenty of opportunities for almost all skills in any settlement, and consequences to each situation.
No uberpowered characters for me, I prefer you either be specialised in a few skills and ignorant of everything else, or be a jack of all trades and master of none who fails as many times as he succeeds and pays the price for that failure. Arcanum progression was about right in my opinion. No QTEs obviously.
That's how i'd do it as a GM anyway, whether you could bring such a system over to CRPGs that's another matter, still copy the basic model of the first two Fallouts and i'd say it's definitely doable.