Not just to avoid oxidation. The fluid plays a useful role in the cutting process. As the spark occurs, you generate extremely high temperatures in its immediate region which explosively boils a tiny volume of the fluid. It’s not clear to me if this generates a cavitation effect that helps material removal, but it does help disperse the material boiled off the workpiece itself; without this, the spark gap tends to fill with condensed solid material that is the “swarf” from the cutting. For the same reason, it helps if you can keep the fluid moving in the region of the cut although it will work without this.
I have demonstrated the SMEE wire EDM machine at a number of exhibitions over the years and one of our problems was in the fluid handling. The simple sinker-type machine discussed above is not so fussy but when you get into wire eroding, flushing the cutting area with clean, low-conductivity fluid is critical. We used deionised water but because it was designed for “up close and personal” demonstrations, we could never use as powerful a jet directed at the cut as the process demanded. The fluid was filtered, passed through a deioniser, and recirculated. We used water as a dielectric.