For a MOSFET, does capacitive gate current only flow through to the source?

Remember, there are two (major) gate capacitances in a power MOSFET: the gate-source capacitance Cgs, and the gate-drain capacitance, Cgd. If the drain voltage never changes, then yes, all gate current goes into the gate-source capacitance.

However, in power MOSFETs, the drain voltage almost always changes. During turn-on, at the point of switching, the drain voltage in an NFET will start to fall. This discharging of Cgd causes current to flow from the gate to the drain. Depending on drain-source voltage Vds, transistor and driver parameters, the gate-source voltage Vgs may have a small inflection point, a Miller plateau, or wild oscillation.

For a better discussion on the Miller plateau, see this answer

There is capacitance from the gate to every other electrode in the device.

When the gate is charged, some current flows through the source. This terminal is often grounded, especially in a switching application.

Some current flows through the drain. While the gate-drain capacitance is often an order of magnitude smaller than the gate-source capacitance, the effect of the charge in it, because of the voltage swing on the drain when switching, is often an order of magnitude larger than that of the source. This is called the Miller Effect, and causes the long flat area in the gate charge graph, which delays the full device turn-on when current is not supplied fast enough to the gate.

Currents from both drain and source eventually end up at the device driving current into the gate, to complete the circuit.