I think these two posts pretty much for me shows what is the issue with Liberty: Even if you intend to go wide, for the first several eras you will only have the 3-4 cities you have in a Tradition/tall built. Liberty may be an investment in the future, but if that's the case, won't you still be better off starting with Tradition and then only after that switch into Liberty, when your first ~3 cities are up and running and you can support an effective expansion?
How will you get the culture to manage that? You can't finish both trees before Rationalism opens up, and on Deity, you can't delay Rationalism. You pick one, then take one or two policies from a second before Rationalism. Furthermore, Liberty allows you to set up those first 3 expansions much faster, as you can hard-build a settler while waiting for the free one, then fast build another one, and spread like wildfire. Again, you'll need that just to get 4 cities on Deity. With Tradition or Honor, the AI will have ALL the good spots before you even get your third settler, since they start with a free one.
I've had some success with coupled Tradition/Liberty builds and then going very wide, but admittedly I'm not a very competitive player (king/emperor), so I'm not going to claim this is a winning strategy for every setting.
On Deity, I've had equal success with both. You need to meet all the other AIs to start trading lux. Wide empires can't afford to be that picky, so you WANT, ideally, one unique lux per city, but if you can't get that, at least settle cities with copy luxes so you can sell them off for GPT/Happiness. On my last Liberty game with America, I had settled six cities and was first to Industrial. I've never been first to industrial with Tradition, but my key to success was that I used my abundant luxes to make GPT deals with the AI, then used a few of my caravans for growth of my smallest cities.
One of Liberty's BIGGEST strengths, by far, is worker speed, especially with the Pyramids, which is an easy wonder even on Deity. With Tradition, I'll often still be improving tiles well into Industrial, but with Liberty, you actually run out of tiles to improve and can start disbanding workers. To even hope to match that speed, Tradition will have to spend the GPT it saves on building maintenance to field 50% more workers, which is time wasted not building something else.
Liberty's other big strength is momentum. Tradition often has to stop what it's doing to expand or improve, but Liberty doesn't care. You can be building a wonder in your capital and still get a free worker. 1 hammer per city sounds like nothing, but then you realize that just doubled your starting hammers, so you can settle high-food locations and not be worried about low production.
Tradition only really buffs the capital, and while it is a good buff, you don't need a huge capital with Liberty. A single Maritime ally provides more food than Tradition does, and if you get Temple of Artemis (hard but not impossible), you're effectively just as good as Tradition.
Liberty is easily on par with Tradition for best starting tree in the game. The finisher is extremely versatile and can be used however your strategy needs. You can use a GS to get faster tech than Tradition, who won't be getting any GS for probably another two eras. You can rush midgame wonders that Tradition DREAMS of but could never hope to get on Deity, you can found a religion without needing a faith-generating pantheon, you can grab an easy CS ally that will be yours just about the entire rest of the game, you can grab an early Great General without needing Honor, you can pop an Artist and delay your artist guild a whole era, and if you find a use for any of the others, they're there too.
I've also had some success with Liberty/Honor starts if I inted to go warmonger, because even though Liberty is bad for starter, Honor is even worse. Obviously if you go conquest Liberty will help you with happiness (if you connect the cities up), and conquering enemy lands will generally give you many unique luxuries to help with happiness issue of going very wide.
Honor is not bad for starting if you want to warmonger early, and if you split between the two, you'll get the best benefits of neither. You need 3 policies in Liberty to get the faster Settlers, and at that point, you're better off finishing Liberty than getting into Honor, because what the hell good is Honor if you're not getting faster barracks, cheap upgrades, and faster exp? You won't get those before it's time to get Rationalism, and if you'd just stuck with one tree, you'd have the policies to dabble in Aesthetics, Patronage, Commerce, or Exploration.
Honor just plays a completely different game than the other trees, you're not going to settle that many of your own cities, maybe 3 tops, and you'll be relying more on your trade routes for science, but that's okay, because you're going to conquer capitals with juicy, juicy wonders. Your culture is better than the others if you go down the right side first, and on Deity, Honor is the only bonus against barbs you'll ever get. Honor is not a common start, but it's certainly viable with an all-in strategy.
On the bottomline, I agree with OP and many others here: Problem for Liberty is that because wide expansion doesn't happen right from the start but only from medieval era, starting with Liberty over Tradition sets you significantly back for a good part of the game, and I'm not convinced that the late-game gain from Liberty (over tradition) is actually big enough to make up for this? To make this worse, because both Honor and Piety only seem to work as secondary trees (with either Tradition or Liberty), this makes for a very heavy domination of Tradition in the game.
Liberty is actually better at setting up shop quickly, much better than Tradition unless you get obscenely lucky, and because of this, you get a large early game boom, followed by a short lull (which is needed to build up necessary infrastructure), then a second big boom as you expand your empire, and if you're Liberty, you'll want to go to war eventually.
Liberty can be seen as the warmonger tree for civs with mid-game UUs. You CAN go peaceful, but you'll likely have to go warmonger eventually. Honor is for people who want to go to conquer others right away and focus on the warmonger half of the tech tree. With Honor, I've conquered capitals with no losses, so handily, on turn 90, that the AIs sue for peace and give me all their cities.
So, why is Tradition so popular? Because it's so
easy. Tradition is pretty much autopilot for people with bad habits and rigid routines, fittingly enough. Tradition is all about build order optimization and beelining to peaceful victories, and people love that because it's much easier than thinking and working, and if you get stuck going anything but Freedom, your empire suffers. Wide empires can work with Freedom. Order is much less effective with Tall empires. It's also less effective if you're willing to do the work to make your empire great. On high difficulty, Liberty is the only way to get more than 3 cities in all the best spots. Tradition has to deal with sloppy seconds, and rarely gets a good spot for its 3rd. Liberty often gets prime real estate in all 4 of its first cities, is much faster at improving those first cities, and you occasionally find a vast expanse that's good for 6 or 7 cities.
Liberty is nowhere near underpowered. It takes more work, but it's even better than Tradition once you get it going.
at normal game speed. That's a big opportunity cost from your capital. Meanwhile, free aqueducts are free. 
