Based on your description, this is what you are suggesting:
I’m not sure if this is exactly what you meant, but there are two problems with this. The first is that the assigning of 192.168.20.1 to the Fa0/1 interface of the router will not allow the router to receive packets from VLAN 20. Where does it connect to?
Secondly, if you didn’t create the subinterfaces on the Fa0/0 interface of the router and the Fa0/1 interface of the switch was still configured as a trunk port allowing both VLANs 10 and 20, then all the frames exiting Fa0/1 of the switch would be dropped by Fa0/0 of R1 because they include a VLAN tag. The interface on the router is not configured to understand these, so the link would fail. This is a case where there is a misconfiguration on each end of the link: a trunk on one end and in essence an access port on the other.
Now my suspicion is that you meant that the IP address of Fa0/1 of the SWITCH should be configured as 192.168.20.1. Once again, this would not work because first of all, a switch port configured as a trunk cannot be assigned an IP address. In order to assign an IP address the port must be configured as a routed port (assuming the switch supports this). This however means that it cannot carry multiple VLANs.
So the only way to really configure router on a stick is with subinterfaces. It’s good that you are asking such questions however, because it shows that you are thinking more deeply about what is going on and you want to more fully understand the concepts involved.
I hope this has been helpful!