Intuition behind the asymmetry between invertibility of key and ciphertext in block ciphers
If you had a block cipher with this property, it would necessarily be insecure!

The most basic requirement for a block cipher is to be a pseudorandom function (PRF). In the game that defines security of PRFs, the attacker gets to choose input blocks $x_1, x_2, \ldots$ and gets back responses $y_1, y_2, \ldots$. It should be hard to distinguish whether the $y_i$'s are uniformly random or have the form $y_i = F(k,x_i)$ for a random (secret) key $k$.

Suppose your block cipher was ``key-invertible'' in the way you describe. Here's an attack against the PRF security of the block cipher. Ask for some $x_1$ and get back $y_1$. Now you do this key inversion process and you can learn (a candidate) key $k'$. Then you can make one more query on $x_2$ and get back $y_2$. If $y_2 = F(k',x_2)$ then you guess that you are getting outputs of the PRF; otherwise you guess that you are getting random responses. 

This attack completely breaks PRF security. It assumes that the key-inversion process is perfectly correct --- that is, given $x$ and $y = F(k,x)$ for unknown $k$, you can always compute that exact $k$. Even if you assume the key-inversion procedure is not always correct (e.g., maybe there are two keys that satisfy $F(k,x) = F(k',x)$ so sometimes you get it wrong), It seems likely to me that this attack can be extended.