I disagree that liberty doesn't work for early warring. I think it's better than tradition for it.
If you mean that tradition gets you a tech lead faster, yes, that's true, but that's not an early rush. It takes too long to get a generation ahead on military tech for Deity and most players end up waiting for Xbows.
A true early rush you rush the AI at the same tech level as them as soon as you can--usually composite bowmen but even before in some cases. Liberty is better for this because of quick settlers and an extra hammer per city to build troops with. So your total rate of early troop production can be higher without sacrificing much infrastructure. In liberty a 3-pop city on on a hill with republic can work another hill since you can't afford to grow anyway and have great production. With tradition if you stop to work any mined hills you slow your growth and lose your advantages. Besides the slight hammer advantage and faster expos and workers, there's happiness problems. Even if you can grow a tradition city up and get enough production to produce good troops without sacrificing your early growth (main tradition advantage) growing your core tall given tradition's tenets means too little happiness to conquer very many cities.
A very standard way of doing early rushes is to REX quickly, slap down like 4-6 cities with liberty, build monument, maybe granary, then comps. You can make more then in tradition so the rush is more sustainable. Tradition grows you core well but you're squandering nearly all of its advantages if you rush early and conquer a big puppet empire because very few of the policies benefit more then the first 4 cities. Liberty on the other hand is set up for conquering big by making every city cost you 1 less happiness via meritocracy when you connect it in via roads, giving every new city +1 hammer, and reducing the cultural penalties of the empire. You can annex and build stuff faster.
Obviously honor opener is the best for early rushes then either of them, but it doesn't have enough infrastructure benefits to sustain itself other then the garrison culture and happiness so conquering the entire world with just honor is not reliable. Works with some civs like Huns though.
Most domination players on multiplayer I think start with liberty and mix in honor at the same time. When you do this you get the garrison, xp, and combot bonuses of honor + the culture, happiness, worker speed, and hammer bonuses of liberty. Also going liberty+Pyramids is the only way you can get 1-turn tile improvement repairs meaning you can farm pillages and heal your troops if you bring workers under them on the front-lines. You can also build roads under them in 1 turn. It helps a lot.
Good luck conquering a lot of early cities and not falling behind without liberty or honor--it's much harder due to happiness shortages and missing the 50% xp bonus. With both you get effectively 2 free happiness just from your garrison and road, and +3 culture. Stuff you were probably gonna do anyway.
You can often have enough happiness and strength between the two to conquer the entire world, capitals and satellites, on standard/pangaea far before tradition reaches artillery.
Tradition/honor works ok but has less of a synergy in my opinion and certainly isn't as set up for ruling that empire which is why most players end up sniping capitals in rapid succession rather then really conquering the world.
Tradition is only good for conquering late-game in my opinion. It doesn't have the sustain to take a lot of early cities. Basically the way to conquer with tradition is stay small and tech till you get ahead on tech and get to artillery then start sniping capitals.
Sometimes you get lucky though...
