why can't we just put R1 between the button and GND
It's called pull-down resistor. In that case switch would need to be connected to Vcc instead of GND.
EDIT: I mean between the right side of the switch and GND. If you mean "between" as in series with the switch, there is no use for it in this case.
And a more general question: why is this resistor even called "pull-up"? There is nothing to be pulled here.
Yes there is, it pulls up the voltage on the input pin to the Vcc level. Also generally there is no R2 in the MCU as on your schematic. Either pin is freely floating (very high input resistance), or if internal weak pull up is enabled, it rises to Vcc. Some MCUs may also have configurable pull-down.
If R1 was not there, the input pin would always be properly connected to Vcc
And where would you connect the switch in that case? Between Vcc and GND and short the power on each button press?
The thing is, depending on the switch state, input voltage need to be either high or low. If there is no pull-up resistor it would be freely floating when the button is not pressed and would be in undetermined state. Likely changing state rapidly because of the picked up interference. Connecting the resistor to GND would make the schematic not working at all. As voltage on the pin would be the same as GND, button press would have no effect.