Apocalypse World: It's very difficult to become a leader without being a very good fighter. The Chopper and Hardholder classes even have moves letting them use the same stat for exercising authority and kicking ass.
Arkham Horror: You can be appointed an Arkham deputy if you murder enough monsters and seal enough gates.
Ars Magica: Codified in House Tremere, unlike other Houses of Hermes. Even after their apprenticeship, a mage is bound to their former master and can't vote on their own behalf until they win their freedom through a formal Wizard Duel. They can also duel to replace their leaders — within certain limits, since even Tremere acknowledge that power isn't the only gauge of merit.
BattleTech: The Clans work this way, being a Proud Warrior Race. They have a Fantastic Caste System where the warrior caste is very much in charge, and martial skill determines rank and command privileges. Trial by Combat is the legitimate way to air one's grievances, or to overturn political decisions that didn't go your way. Once you obtain a Bloodname and enter the Clan Council, the rules change a bit: No-one is allowed to challenge you for your Bloodname while you still live, and Khans (who are elected by the Clan Council) have to be able to play politics as well. Being nominated by a member of the Council also makes it much easier to be able to fight for a Bloodname in the first place. Generally speaking the less successful Clans (Smoke Jaguar and Ice Hellion) tend to play this trope more straight while more powerful and thriving ones (Wolf, Jade Falcon, Ghost Bear, Diamond Shark, and Star Adder) elect leaders based off both combat ability and political prowess.
Deconstructed in a very subtle fashion, rarely explicitly stated. The Clan system selects leaders based on skill with single combat and in-Clan politicking and leadership ability. It does not select for unit-level tactics, strategical thinking, logistics, economics and civilian administrative skills. The Smoke Jaguars exemplified this weakness: Their leadership was so focused on fighting that they neglected every other aspect of their Clan. Lincoln Osis, the final Khan of the Clan, was a skilled tactician but an incompetent strategist. His total neglect of his Clan's industrial production or logistics meant that his Clan was never able to recover from the damage inflicted during the Clans' invasion of the Inner Sphere and when the Inner Sphere decided to attack the Smoke Jaguars several years later, the Clan was completely wiped out.
Most depictions of devils work on a very orderly version of this. Demons are similar, but drop the "orderly" part — as a species dedicated to selfishness and chaos, the only authority demons consistently respond to is "they can kill me if I don't follow them".
Dracotaurs respect nothing but strength and power, and their tribes are invariably led by their strongest warriors or magic-users.
The drow that follow the spider goddess Lolth, which includes a great majority of the entire race. There are lots of rules in their society that demand severe punishments for betrayal and murder, but only the victims or their immediate families can bring a case before the high priestesses. Thus the only way to power is to kill everyone in your way and intimidate everyone who could make your deeds publicly known. Everything is legitimate, as long as you can get away with it.
The Ordning as it applies to frost giants is based on pure physical might, with rank determined by challenging other giants to things like wrestling bouts (you get their rank if you win, and they have to take yours).
Applies to the system as a whole. Since all abilities, from ability to fight to ability to weave baskets, are based on level, anyone that is exceptionally good at something, even a purely non violent profession or trade, is liable to be able to take at least a small army on by themselves... and win. As there is no shortage of ambition and predators — literal and figurative — those who keep a position of authority are those with enough personal power to kick the ass of anyone who wants to take their place. Often ends up being recursive with Authority equals Asskicking, both in that getting to the position in the first place is liable to improve your combat abilities (even if you do not fight your way there) and that it is often necessary to be proactive in order to maintain your position.
Forgotten Realms frequently shows how such things happen. Let's take Wyvern's Spur and a story of one ex-sellsword, now the King's governor:
Giogioni Wyvernspur: Is that a prerequisite for your job? Samtavan Sudacar: Got to make a name for yourself at court. Slew a frost giant that was terrorizing merchants in Gnoll Pass. That's how I got into politics — a service like that has to be recognized officially.
Exalted has this going on with the Yozi Cecelyne, Hell's lawmaker. Her rules stem from the belief that the strong are to rule over the weak. Her other laws are often arbitrary and sometimes outright contradictory, but this is the one truth she holds to absolutely. Indeed, it's one of her unbreakable themes as a Primordial being, so she can't even conceive of another way it could be. The random contradictions in the other laws are mostly because she is really pissed off at her current state and is taking it out on Hell's denizens.
The Vikings of Fate of the Norns: Ragnarok live in a time when every little problem or power struggle escalates very quickly. As a result, people in power have to be able to kick a lot of ass if they want to stay (case in point: King Erik BloodAxe murdered almost every single one of his brothers to keep his place).
Numenera: The people of the underwater city of Ahmas firmly believe this: in their society, might and combat prowess are the only ways to achieve status and power. Their massive gladiatorial games are the closest thing they have to a formalized way to climb the social ladder, and their king is thus the best and deadliest fighter of their lot.
Chaos Lords gain their position through sheer power and favor from the Gods. The latter is gained by the Chaos Lord's quality in the former. Khârn the Betrayer, for example, is the greatest of all Khorne's Champions due to his phenomenal fighting ability. When a Chaos Lord is slain, the best of the warband usually take each other on until only the strongest remain, and then the strongest fight each other so that only the greatest Champion remains.
Ork society. "Da bigga an ork is, da more dat 'es da boss." It actually goes both ways — orks know who is their boss because the boss is bigger, but orks actually grow as a response to rising in the social hierarchy-the bosses really are bigger because you start growing when you become the boss. Orks find human social hierarchy very confusing because humans "are all 'bout da same size" and wonder how they ever decide who the boss is (some resort to picking either those who lack the helmet/hat or have the fanciest among other 'ummies which usually works). Ghazghkull Mag Uruk Thraka, the greatest of all Ork warlords, has a head the size of a human torso, and arms thicker than a literal tree trunk. He's actually closer to the size of a large Killa Kan or small Dreadnaught than a typical ork. Nothing more needs to be said about his strength, or his love of all things dakka. And he's just the "heir" to The Beast, the greatest Warboss(es) that ever existed. In terms of asskicking each one was a hab block-sized giant with tusks like trees, that personally destroyed entire armies and were a match for Primarchs in a one-on-one fight. In authority, they commanded essentially every ork. Not even Horus came as close to destroying the Imperium as the Beast's Waaagh did.
The T'au have a more formalized version of this. The only way to receive a promotion is to complete a four-year tour of duty and than succeed at a Trial by Fire. So not only are unit captains the ones who have already excelled at their battlefield role, every Shas'O has spent a decade or two kicking ass all the way up the ladder.
Abaddon the Despoiler is the Warmaster of Chaos mainly because he can literally tear apart any and all challengers to the position. However, he's a bit of a subversion as well, since he's not actually that great a commander: on the tabletop he has incredible close-combat stats but barely buffs his forces. Great warriors don't always make great generals. This has changed as of 8th edition — he's still a beast of a fighter but he actually buffs his troops too.
Every seven years, the incumbent Supreme Patriarch of the Imperial Colleges of Magic can be challenged to a Wizard Duel for the position by any other Lord Magister.
Ogres work this way. The only requirement to becoming tyrant of a particular tribe is to beat, and then eat, the old one.
Orcs warbands are led by the 'ardest and toughest of their number. An Orc who wants to become the new Warboss needs to challenge, fight and kill the old one.
Beastmen recognize only three things as conferring authority: the favor of the Dark Gods, the size of one's horns, and skill in combat. The most fearsome warriors rise quickly through the tribe's ranks, and the title of Beastlord is kept by whoever is tough enough to defend it against a constant stream of challengers.
When he awoke after the brain transplant surgery that turned him into a cyborg rat monster, the former Norse Berserker Ghoritch made a mad dash for freedom, and in the process slaughtered his way through the tides of monsters that his creator sent after him. The lords of Clan Moulder were so impressed by this display of carnage and power, alongside his intellect and strategic sense, that they made him leader of their armed forces.
