They're not talking about the stars, but the hexagonal spikes sticking out of the stars. Those are artifacts of the hexagonal lenses that the telescope uses.
I think you're both right to a degree? They are lens artifacts and we can't just remove them because we don't have the data of what's behind the spikes, as that is what the sensor can see. We'd be filling them in with black or our best guess based on neighboring pixels, either way it wouldn't be what one would "really" see. Unless we composited images from another source, then that'd come closer to accurate.
That seems right. For popular consumption it'd probably be nice to have a black-filled version or something, whatever is least visually off-putting, but for science that seems much, much worse.
No, the hexagonal spikes are artifacts of the arms holding the secondary mirror in place.