Things are not so simple: how do you key the cipher? with the same randomness as the input string? If so, then that input distribution could be a bad one for the cipher, in that it might fail to generate a sufficiently scrambled output.
This can be mitigated if you use a unkeyed hash function instead, but even for these we don't have any provable guarantees unless you assume the function is a random oracle, which is an ideal object that cannot exist.
Yeah, if we’re starting with the “1 in a million” unfair coin then most of the input strings are going to be all 0 or all 1 which means the cipher is going to give the same sequence every time. Not useful!
This can be mitigated if you use a unkeyed hash function instead, but even for these we don't have any provable guarantees unless you assume the function is a random oracle, which is an ideal object that cannot exist.