Why doesn't this capacitor short this circuit?
The capacitor is in fact a short circuit, however only temporarily.

When you first turn on the power supply, the capacitor will act like a short circuit during this initial transient phase. There will be a large inrush current as the capacitor charges up (*).

Similarly, if you take a fully discharged capacitor, and connect it to a power supply which is already at some DC voltage, you will get a sudden inrush current into the capacitor which if large enough (e.g. very large capacitor) can be enough to overload the power supply. This is especially problematic if switching a large capacitive load into a circuit.



Once the power supply reaches its steady-state DC voltage, and the capacitor charged up to the same voltage, the capacitor will no longer act as a short circuit on the supply because it cannot conduct a DC current as explained by the other answers.

If however you have some noise or ripple signal, or a voltage transient, the capacitor will again act like a low impedance or short circuit (frequency dependent) for this noise signal or voltage ripple. This is how the capacitor has the effect of smoothing out the power supply.



(*) The power supply may provide some form of inrush current limiting or controlled voltage ramp-up, in which case the current flow will be limited.