And is liberty definitely better for wide?. I boringly pick tradition almost every game, the variation in my game is my second policy tree (usually piety, but with raging barbarians i dip into honour, and on water maps dip into exploration)...3rd tree is always rationalism.
Yes. Just look at the policy tree
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It unlocks Pyramids which if you can get it done and then combine it with Citizenship, your workers are going to rock all over the place, roads and improvements built in no time. Unlocking also provide 1 culture per city. Republic and Meritocracy are always useful, especially early on. And finally Representation comes as a proper finisher, reducing culture cost for the rest of newly founded cities, puts you in the golden age and you get to choose your great person.
I would also like to add that Liberty goes well with Tradition, you just need to figure it out how to combine policies because you want to finish both as soon as possible.
You forgot the most vital policy in Liberty: Collective Rule. Basically, the strategy is to slowly build up your capitol while you rush Collective Rule, then start churning out settlers like nuts to settle as many cities as you can. Collective Rule not only gives you a free settler, but the increased settler production lets you essentially get one free settler for every 2 you build, which is why it's better to wait until you unlock it before pumping out settlers.
In fact, Collective Rule is so powerful that it's actually used for Tradition/Liberty mix strategies, which go Tradition opener into Collective Rule into finishing Tradition.
You will definitely be behind on your NC timing and you definitely won't be science leader unless you're Maya or China, but you will be hammer leader, which means that you might not be able to get Universities quickly, but you will be able to get to Crossbows before other people and then steamroll your neighbor with a force of 7-8+ crossbows and a capture unit.
The biggest things Liberty tends to suffer with are early gold, early happiness, early border growth, and early science. Early religion can be vital to combating all 3 drawbacks, with gold coming from the Founder belief, Happiness coming from follower beliefs, and science coming from the Reformation belief (Jesuit Education or Glory to God, the latter will let you purchase GS without having to complete Rationalism and GE without having to complete Tradition). Policy points can also help nab some key stuff that Wide may wish to rely on, eg. Wagon Trains in addition to Mercantilism, the four policy points in Piety, etc. Their biggest advantage is early hammers, usually to produce military units. As a result, civs with UB's and/or UA's that address those drawbacks and/or UU's that can capitalize on early hammers are best suited for wide Liberty.
Examples of good Liberty civs or civs more suited to Liberty than Tradition include: Arabia (UB helps with early gold, great UU to spam), Byzantium (UU is decent, UA can help a lot if you get a good religion), Carthage (free harbors is best when you have a lot of cities that can capitalize on it), Celts (UA helps with religion, UB helps with happiness), China (UB helps with science and gold, UU is great to spam), Egypt (UB helps with happiness and gold, UU is great to spam, UA helps nab key wide wonders like Sistine), England (two great UU's to spam, free spy helps with early science), Ethiopia (UB helps a lot with early religion), Huns (two great UU's to spam, free tech helps with early science, free production is flat so it's better utilized by wide), Inca (UA helps with early gold and military movement during pushes, UI is just obscene), Iroquois (UA and UB are both more useful for wide than for tall), Maya (UB helps with early science and early religion, UA lets you get GP's without have to work specialist slots, which are harder to work for wide anyway), Mongolia (UU is great to spam), Persia (UB helps with early happiness, UA helps with military movement during pushes), Poland (UA helps with everything, UB helps with early gold), Russia (flat production boost is utilized better on wide than on tall), Shoshone (UA helps with early border growth, first UU helps with everything, second UU helps for Industrial pushes), Songhai (UA helps a bit with early gold, UB helps with early gold, UU is great to spam), and Spain (UA helps with everything if you're lucky). Notice I did not mention Rome because % hammer modifiers are not as significant for wide as they are for tall, and neither UU is really that ideal to spam (maybe Legions, but you're sacrificing a lot tech-wise if you go for early Legions vs. early Markets or early Colosseums).