Is K=1 a valid random integer in ElGamal signature?
Why $K=1$ is not excluded? You can also ask why $K=2$ is not excluded. It always possible to calculate $g^2$ and by keeping $(g^2,2)$ in a lookup table, you can also recover the secret key if you see $S_1 = g^2$, in which case $K$ happen to be 2. So as $K =3$, $K= 4$ and so on so forth. As you can see, 1 is not that special. 

The reason why they are not excluded is because that $K$ is chosen at uniformly random from the range. For an adversary who has only limited computational capacity, it can only guess polynomial number of $K$ and does the computation, but if the signer picks his $K$ at uniformly random, the probability of the adversary guessing the right $K$ is negligible. So as long as the signer does his job properly, we don't need to worry about $K$ being a specific value.