In Belgium, you are allowed to do this. When I moved to Copenhagen I was seriously annoyed for the reasons you mentioned when I realized I had to first cross, wait and only then go on with my ride. However, I started realizing that the Copenhagen way actually works very well, and it is a LOT safer, and so much more relaxed (not slower). Merging into traffic and crossing the street to get to the lane to turn left can be a stressful affair. I used to bike in Brussels and I can assure you that is a jungle over there: you have to be reckless as a biker (and so I was) if you want to get anywhere at all. On the upside: a fast morning ride through Brussels from home to work was one big adrenaline shot. Once behind my desk, I was awake, sharp and ready to start working :-)

Additionally, cars going for a left turn usually have a separate traffic light which is red when the main direction has green. The time you gain using that lane compared to the "box turn" is minimal as such.

A lot is also context: Copenhagen has a very extensive bicycle lane network that does not involve any cars that is constantly expanding. Cykelslangen is a nice example of such a bike only short cut https://www.youtube.com/watch?v=DmKXYZZy6dI, http://www.dw.dk/cykelslangen/, http://classiccopenhagen.blogspot.dk/2014/06/cykelslangen.ht.... When going around the city, you are just faster with a bike because you take more direct routes (away from car lanes) with less traffic lights. If I have to take a left turn on big busy car street, I couldn't car less if I had to wait for making a box turn: I am faster anyway, and the properly tuned and fast switching red-green lights minimize the wait.