Kelly can report anything that she pleases. This has nothing to do with the Constitution.
The Constitution doesn’t guarantee anyone the right to moderate a debate.
If debates were run by a non-media entity, then the press connection is more tenuous, and your position would be stronger.
Her job as a moderator is because of her position in the press. The debates are run by media outlets, aka The Press, with reporters as moderators, so yes, it IS part of her job as a reporter.