The specific effect of the Liberty opener is "founding a new city with an Amphitheater has no effect on policy rate."
Except of course, that this isn't true.
Suppose that my next policy costs 3000 culture.
I have a supercity with 100 culture per turn because of stacked landmarks, wonders, etc., and three other cities with 20 culture per turn.
So it will take (3000/160 =18.75) turns to get my next policy.
Founding another city with this policy active will increase policy costs to 3,600 (I think?), and this city with monument and ampitheater will generate 10 culture per turn. Now it will take (3,600/170 = 21.17 turns to get the next policy.
I don't have a problem with the policy effect or the simplifying language, but recognize that it is only literally true under a whole some particular assumptions. It isn't true generally.
I like getting the culture bonus out of the way right away so that anxiety is gone.
Ok, but not everyone has to play like that. There *should* be interesting tension in tradeoffs; if I get the reduced policy cost policy now then I will get the policies after that faster, vs if I get the other policy now I will get the useful effects now but will be slower in generating later policies.
The effect also rewards development more than +2 culture per city
It reduces the effect of building cities on getting more policies; it doesn't reward other kinds of development, like tile acquisition.
I also kind of like how the 3 openers provides culture in a very different way:
Killing units
Happiness
Developing cities
If by developing cities you mean "building more cities", the +2 culture per city does this too; it gives a larger benefit when you have more cities.
The instant culture buildings effect lets people who research Drama immediately build 4 amphitheaters in their first 4 cities, and start construction on a National Epic right away. This is a powerful and exciting strategy
I don't find it powerful or exciting. And to do this I would have to get 4 cities up in my slow-to-found-new-cities Tall empire *with* monuments built before I built an ampitheater in my first couple of cities. I'd rather build the ampitheaters earlier.
I don't find it exciting for a policy to just give me something I could build on my own.