Why is the result zero when I compute the linearization with respect to a variable time delay?

The reason why the answer is zero is because of the specific way the "Variable Time Delay" block is treated during linearization. The input to this block is not meant to be used as "Input perturbation" and as a result the linearization is not accurate. For an alternative implementation, please look at the attached model. As opposed to using the "Variable Time Delay" block, one could use the "Clock" block and add or subtract the delay time as necessary. In this approach, when the variable time delay is set as "Input perturbation", the result is the expected one. The example models the linearization of "y = sin(t-t0)" where "t0" is the value of the variable time delay.