M4 is diode connected. If M4 directly connected to R than V(R)=(Vsupply-Vth(M4)). V(R) varies with Vsupply ==> I2 varies with Vsupply. What we want is constant current independent from Vsupply. To accomplish this we use M2 to isolate R from diode connected M4 therefore we can set voltage across V(R)=Vth(M1).
V (R) = V (GS1) can not be enforced. Since gate of M1 is high impedance and can not enforce a voltage on R. V(R) is solely controlled by V (R) = VDD-V (SG4 ) and so V(R) follows Vsupply variations.
In the original circuit with M2 not removed, through feedback V (R) = V (GS1) is enforced. Since gate of M1 in that case senses the V(R) and through M2 enforces a voltage of V (R) = V (GS1). Since M2 is in place between M4 and R as an impedance transformer (common gate configuration) it isolates M4 and R and lets enforcing through feedback V(R)=V(GS1) which is independent of Vsupply variation at first order.