tl;dr: for reasonable values of R and C, both OUT and DISCH have enough drive capability to handle discharging / charging the capacitor.
Normally, you’ll see 555 astables use the DISCH pin, and an external resistor pulls up that pin to charge the capacitor. This is because the discharge pin is open collector with no high drive of its own.
When the OUT pin is used instead to achieve 50% duty, it has its own high drive so it doesn’t need a pull up resistor. So the capacitor charges up through R2.
R2 has to be a high enough value, or C2 small enough that the charge/discharge current does not overwhelm the OUT pin drive capability.
(Ed. Note: R1 isn’t necessary.)
Here is a simulation of the 555, showing its internals in the ‘typical’ astable configuration using DISCH (simulate it here):
This astable relies on the pull-up resistor to charge the capacitor. It isn't capable of 50% duty cycle because the charge and discharge resistances aren't equal: charge is 11k, discharge is 10k when DISCH grounds the pull-up.
We can fix this by reconnecting the 10k resistor directly to OUT (simulate it here):
Here, charge and discharge are both through the 10k to OUT. The the charge/discharge currents are equal, so the duty is 50%. Yet the current is small enough (0.33mA max) that OUT can handle it without losing its state.