Is a VOR check required to legally fly an ILS approach under IFR?
This regulation does not apply to an ILS approach.

You are right is saying that "the VOR system of radio navigation" refers to actual VORs.

If the flight is conducted using GPS to an ILS approach then you aren't using VOR for navigation. It does not matter that ILS uses the same instrument display.

Intuitively, though ILS and VOR use the same display and frequency band, they are functionally different. The VOR beacon sends out a omnidirectional signal, from which a bearing (0-359 degrees) is calculated. The ILS sends out a focused, directional signal, which the receiver centers on.