Bypass capacitors aren't really used to prevent a whole-system voltage sag - they are there to deal with the effects of high-frequency signal switching.

The traces supplying power aren't theoretical, so they have a nonzero impedance. This means that when there is any change in power demand due to transistors switching, the trace will try to resist this change in supply current. An increased power demand leads to a very local voltage drop, a reduced power demand leads to a very local voltage spike.

Bypass capacitors are placed as close as possible to the individual chips to reduce the impedance between the capacitor and the chip. You even intentionally use smaller capacitors because the bigger ones have a higher impedance.