If the openers of the basic trees provide culture (they should), the way they provide it needs to be an active one. The problem with Tradition is that you don't lose anything practically by chosing it as a first policy, but it speeds up your policy pick-up rate considerably. All three* should be similar in strength and require action. Honor killing Barbs, Liberty founding cities and/or building settlers/workers and Honor maybe building Buildings or growing your capital. Depends a bit on how one can balance them to each other.
*If we include Piety as a basic tree, that still holds true. Here, the culture comes from the Pantheon belief chosen, and if one is chosen without culture, then that's a considered decision by the player to foresake policy pick-up rate for more x,y or z.
Though to be fair, the above liberty suggestion wouldn't be faster than the current +1 culture, since it takes more time to build settlers than to get policies 2-5. Thus, one could argue that a hammer bonus to cities works with liberty (new cities need every hammer to get up and running) and a food bonus in tradition (growing population in one city is needed in order to get science and wonders). But if you then put a culture policy in the tree, it gets kinda to be the next logical pick anyways (spend culture to get culture, if you do it early, you get the most and over time, more from the other policies...)
Thus, one proposal:
1) Move the "helps getting culture-policy" to a National Wonder unlocked by the opener of a Tree (Victory Column, Forum Romanum, Monumental Grave - Needs better names).
2) These don't cost too much, but take up valuable time you could instead use on building a scout, worker or Pyramids. Maybe we include food in their construction (similar to settlers) to not punish hammer-poor start areas.
3) You can only build one of them, but still are allowed to open up the other trees. This way, one can make them strong enough without imbalance by someone taking all openers.
4) Piety still wouldn't be affected since a culture-panthen then would be the only way to go "culture heavy" early on, no?
EDIT, since there are new posts now

: One advantage I see with the above is that the openers then can provide a choice (food, production or gold), but of course it's one more item on the building list which is already choke full in the beginning (but it could really only take 2 turns or so to build...). I really like the idea of losing culture on openers since it seems to make it easier to balance.