Well, my main criticism is that Bohr's nearly esoteric ideas lead still today to some more esoterics, as can be seen in this subforum of PF. Instead of discussing how QT is really used in physics, we still discuss old pseudo-problems of interpretation. For me Copenhagen is just fine, as soon as you just strip off the nonsensical and unnecessary collapse idea, which finally leads to the minimal interpretation, which is without any open physical problems in the realm, where QT is formulated today. The real physics problem is that there is no comprehensive "theory of everything". We still only have an effective description of space-time in terms of GR (which is basically a classical relativistic field theory of the gravitational interaction) and all the (known) rest as relativistic local QFT (in terms of the Standard Model). That's the true problem, not some fictitious "measurement problem".
Another thing is "complementarity", which is Bohr's invention, and as far as I know from reading books on the history of physics that's the one idea Bohr was most proud of. What the heck did he want to really say? I think it's the simple fact that the result of a measurement depends on which observable I measure and how I prepare the object I'm measuring. The standard example is the double-slit experiment:
If I want to resolve through which slit each particle has gone, I cannot get an interference pattern for the particle distribution on the screen when using an ensemble. If I want this interference pattern I can't know through which slit each particle has gone. Complementarity seems to say that this is what has been called the "wave-particle duality" of "old QT".
I think, however, that this is misleading and "new QT" has resolved this riddle about "wave-particle duality" precisely because of Born's probabilistic interpretation.
For the double slit I consider this to be clearly seen as follows: First of all to make sense of saying that a particle has gone through the one or the other slit, it has to be prepared to be sufficiently well localized when arriving at the slits to begin with. This implies that the wave function is a wave packet with a width that it much smaller than the distance of the slits. If it then goes through one of the slits at all, it's possible to say through which slit it came, by putting the photoplate (CCD screen) for particle detection close enough to the double slits. Then the spread of the wave function is still small enough to be able to localize the particle sufficiently such that the position of the particle behind the slit reflects through which slit the particle came.
However, with the very same setup you can also get interference, if you only put the photoplate far enough from the slits, because then the partial waves for the particle running through the one or the other slit have become broad enough to overlap and to interfere. There's no wave-particle duality nor some strange "complementarity": It's simply the broadening of the free-particle wave function leading to interference of partial waves going through one or the other slit. Yet you cannot interpret the particles as "being the waves" since each individual particle leaves only one spot on the screen (no matter whether in the "near-slit" or the "far-slit setup"). This brought Born to his probability interpretation.
Particularly, this is all consistent because of the probability interpretation and the Heisenberg uncertainty relation following from this probability interpretation: The narrower the slits are, the faster the partial waves overlap and the closer you must put your photoplate to the detector to still resolve through which slit the particle has come, i.e., the better localized the particles, that came through one of the slits at all, the more uncertain the momentum becomes and the faster the partial waves thus overlap.
The conclusion is that all pseudo problems vanish by simply accepting the probability interpretation. There's no need for "wave-particle dualism" or "complementarity" or any other esoteric philosophical ideas but only the profound conclusion from the study of quantum phenomena for the last 119 years that nature behaves, at least as far as we can observe phenomena, is intrinsically probabistic.