I wouldn’t say it “runs on” virtual particles per se. I think the virtual particle terms are more tracking the interactions between different fields. I would say it runs on quantum fields on a curved spacetime, yeah. And, as for what precisely a quantum field is, this is somewhat mysterious, but generally it is a quantum version of a classical field, where there is a value (e.g. “value of the electromagnetic field”) at each point in spacetime. For quantum fields, instead of each point having a definite value, for any region there is an observable for the total value in that region.

As for how the curvature of spacetime fits with all that, that is an open question that has yet to be resolved. Well, constructing a quantum field theory within a given curved spacetime is fine, but we don’t know how exactly GR and QFT fit together.

I expect that your response is going to be to call these “abstractions” or something, as if this does anything more to discredit them than complaining that any idea is “just an idea”. But these are measurable things. That which can be measured is a real thing.