Garnet has a history of not making good choices, so it makes sense to her character that she stayed with the party, even though yes, she really should have stayed behind. Would the people of Alexandria really noticed that she was in terrible shape? It seems to me like don't see her very often anyway, since she spends so much time running around with Zidane, so she could have easily stayed out of the public view and recovered with very few the wiser. Or, if being in the castle wouldn't work, there's surely some retreat she could have gone to recover. How many rulers of any country don't have some place away from their seat of power to go and have some r/r?