Does a key derivation function always generate an even distribution of bits in the resulting key?
It will be uniformly random for such a simple statistical test. The problem is that you are treating the probability of bits having a particular state as independent of each other. You would need to look at the conditional probability distributions of certain bits being set given other bits.

The entire joint distribution of output and input bits for a function that maps $\{0,1\}^a \to \{0,1\}^b$ requires on the order of $2^{a+b}$ memory to store. Clearly it becomes prohibitive to store and sample it for even a small function that maps 32-bits onto another 32-bits through a fixed transformation (i.e., without including any key bits that define a particular permutation).

The full conditional distribution of a function completely determines it's output. It can be seen as a transition probability matrix in a Markov model. Because of this, the distribution will encode any bias of the function, such as a preference to produce certain bit patterns (with probability).