Here is a thought-experiment to show the importance of displacement current through a capacitor in the real world.
1. Set up two parallel plates, each 10 cm by 10 cm, spaced by 1 cm, inside a transparent jar that can contain the sand discussed below.
2. With air between the plates, k = 1 (dielectric constant), the capacitance will be approximately 0.9 pF. (Even without a material dielectric, there exists a polarization field between the plates. When we add material below, the polarization increases for a given applied voltage.)
3. Connect an AC generator (output impedance 50 ohms) through this capacitor to a 50 ohm load and AC voltmeter. The total circuit impedance is 100 ohms (resistive) plus the reactance of the capacitor.
4. At 10 MHz, the reactance of the capacitor is about 18 k and dominates the total impedance of the circuit.
5. Now, start adding SiO2 powder (sand) to the region between the plates. It is a good insulator, but solid SiO2 has k between 3.7 and 3.9. Since the powder has a lower density than the solid, assume k = 2.5. As the sand level increases from zero to fill the gap, the capacitance increases from 0.9 pF to 2.2 pF, and the current in the circuit (measured by the voltage across the 50 ohm load) will increase by almost the same factor as the reactance decreases.
With these parameter values, the voltage across the capacitor itself is almost constant (since its reactance is much larger than the total resistance of 100 ohms), but the displacement current increases with the increased polarization in the dielectric. Current continuity includes the conduction current through the two resistors and the displacement current through the capacitor.
Moral: just as there are many flavors of ice cream, displacement current is one type of the general phenomenon known as "current", along with ionic current, conduction current, beam current, etc.