Why do we choose the operating point from the $I_d$ vs $V_{gs}$ plot instead of the $I_d$ vs $V_{ds}$ characteristic (similar to what we did with the bipolar transistor?) Can we use $I_d$ vs $V_{ds}$ instead?

Yes, you can use the $I_d$ vs $V_{ds}$ plot to find the operating point. You can use the right-hand part of the circuit Kirchoff Loop equation to write $I_d = \frac{V_{dd}-V_{ds}}{R_s+R_d}$, draw it as a load line and find the intersection with the graph like Eggleston Fig. 5.10. This will give you $I_d$ for a given $V_{gs}$ in the saturation regime. You would then find $V_g$ for this $I_d$ and $V_{gs}$, from $I_d= \frac{V_g -V_{gs}}{R_s}$, and choose resistors for that $V_g$ from the voltage divider equation for the left hand side of the circuit.