Diode circuit analysis with constant voltage drop model
My question is ,Why Diode On voltage is the only cause for the current
flow through R_1 ?

If you are modeling the forward conduction of the diode as a constant voltage then there is also a constant voltage across R1. With a constant voltage across R1, due to ohm's law the current through R1 is constant.

Does V_1 not have any role on causing current flow through R_1 since
V1 is in magnitude greater than Diode_On voltage?

All V1 does when it alters is change the current through D1 but, while ever the voltage is constant across the diode (forward conduction) the current through R1 is fixed by the diode forward voltage.