In planar 2R the singularity condition involves the second joint variable. In spatial 3R, singularities involve second and third link lengths and angles. Why does the first joint parameter never feature in the singularity condition?
Singularity is related to the shape of the arm, not its orientation.
Consider an RR arm. It is singular if the two links are parallel: either fully outstretched or folded back on itself. If the arm is straight, a function of $\theta_2$, then it doesn't matter what $\theta_1$ is, it will still be a straight arm, just pointing in a different direction. You can extend this argument to additional DOF.