Z value is not arbitrary, it is a hash produced from the transaction.
You also can't produce a valid signature (r,s) with the public key, otherwise you could steal any bitcoin that has ever existed by just having the corresponding public key!
What you are saying is correct.
But I am right.
I can create a valid signature RSZ value with only the public key, but
I am writing here because I do not know how to make a raw transaction in reverse.