There are two things should be random in SSS. The coefficients of the polynomial. And the x-values where the polynomial is evaluated.
This is false. Where did this idea come from?
Non-random x values is probably relatively benign, since all information about the polynomial is in the f(x) values. Wikipedia does not even mention that the x values should be random.
What does "probably relatively benign" mean in cryptographic terms?
Pseudo-random coefficients is certainly a bad idea.
Why is this?