is the perfect secrecy of one time pad dependent on the xor randomness preserving property?
(Answering my own question).

One time pad retains perfect secrecy using any operator $\Delta$ showing the following property: for a plaintext $M$ and a key $K$, the cyphetexts $C =  M \Delta K$ is such that there is a the same constant number of $K$s that encrypt $M$ to $C$, for any tuple ($M$, $C$). Specifically, if the length of the key equals to the length of the plaintext, then for each pair ($M$, $C$) there is a single, unique key that encrypts $M$ to $C$.

Therefore, not only xor but also other operators, such as applying Vigenère cipher with a uniformly generated key whose length equals to the message length, are suitable for one time pad.

the "random preserving" property, i.e. the uniform distribution of $C$ given a uniform distribution of $K$ and regardless of the distribution of $M$, is indeed equivalent to the above definition: if $\Delta$ preserves randomness there is necessarily the same number of $K$s for any pair ($M$, $C$) - otherwise, the distribution of $C$ wouldn't have been uniform; and if there is a constant number of keys for any pair then necessarily all cyphetexts are generated with an equal probability.