You are correct that R1 alone could work (with some tuning), but it's fiddler.
I'm assuming the voltage at node 3 is the output of this amplifier.
If you only have R1
You have to tune R1's resistance so that the output of this amplifier (when there is no AC signal on the input) is approximately halfway between the voltage rails. This gives your output voltage lots of space to move before it clips.
If R1 is too small then the transistor will turn on too much and output a low DC voltage, giving it little room before it clips. Similarly if R1 is too big then the transistor will not turn on enough, so the output voltage will be near the positive rail and give your output less room to move before it clips.
Now consider that the beta/HFE of an NPN changes between each transistor, ambient temperature and the value of the resistors on the output. There is no one value of R1 that will always work in this circuit, you would have to connect a pot to tune it for every one you make (and maybe attach a heatsink to it).
If you only have both R1 and R2
R1 and R2 divide the input voltage to be half of the supply voltage. Your transistor's Vbc will be perhaps a few volts, so this puts the output voltage somewhere near the middle too. Changing the beta/HFE or temperature of the transistor won't change this too much. You could even swap it out for different models of transistor and it will still probably be tuned well enough to work.
(In reality I'd choose slightly different values for R1 and R2 and I wouldn't exactly centre the output voltage because that's not ideal for distortion in this style of common-emitter amp, but hey it's probably close enough).