Maybe safer but also a way to tell cyclists to be very patient and don't think of going anywhere quickly. Cycling on sidewalks is only a little worse than that.
I prefer doing like cars, waiting in the middle of the road. There are usually cars waiting there too and they act as a sort of shield. Obviously on large roads it can be difficult to make it to the center of the road and in that case I do a Copenaghen left myself.
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.
I agree the quick light cycles are a big help, also for pedestrians. You rarely wait more than 15-30 seconds on the other side for the light to change, so it's not really much of a delay. I do find the really long light cycles in many American cities annoying as a pedestrian.
Another reason few people in Copenhagen mind the "Copenhagen left" is that since cycle lanes are usually physically separated from the car lanes, it doesn't even occur to anyone that entering the car lanes to turn would be an option. You'd have to physically cross the barrier separating the bike and the car lanes (sometimes a row of parked cars, sometimes a curb, occasionally bollards), which would not be an obvious or convenient thing to do. That part of the arrangement also seems much safer to me. I don't like riding in a bike lane that is essentially just a repainted shoulder, lacking any means to physically exclude cars from the lane.
I prefer doing like cars, waiting in the middle of the road. There are usually cars waiting there too and they act as a sort of shield. Obviously on large roads it can be difficult to make it to the center of the road and in that case I do a Copenaghen left myself.