Well, if you eliminate the position but then start looking for someone to fill it, that could cause trouble. Why? Because the reason for separation was "position eliminiation" but the position clearly wasn't eliminated because the Company immediately started looking for a replacement. That makes it look like the elimination was a pretext for something else. That scenario has played out against the employer as evidence of discriminatory animus.
If the position is truly being eliminated, the only way this could cause a problem is if there is someone else who clearly needs to go but this person was chosen instead.
Although "employment at will" permits termination for no reason, nobody really believes in "no reason". That is, there's always a presumption of a reason. So if you are going to fire someone for "no reason", be prepared to have that decision questioned. For example, you may be asked about what were your reasons for not firing someone else for "no reason", which starts making it hard to have no reason.
If it were me, I'd tell the new supervisor that they need to use the standard disciplinary process to get this person into acceptable performance standards or out of the Company and that the Company would not be taking any shortcuts.