The simplest way (imo) to think about the Oberth effect is "you want to maximize the amount of time you're falling inwards (i.e. speeding up) and minimize the amount of time shooting outwards (i.e. slowing down.)"

It doesn't just work with orbits and rockets and propellants, it works just as well with an oscillating weight on a spring that you flick with your fingers.

It's about ∆v, so I'd say it does regardless of how you got that ∆v. I don't see a reason it couldn't work with solar sails, for example.

If anything, it should increase your inertia, since you're not sacrificing mass for ∆v.