Natures don't perform actions, people do. Christ can do everything proper to both God and to Man because He is one Being (Subject) who is both.
Thus, Christ can perform actions that are proper to God or
to man, even when those actions (like dying) are NOT proper to both. When an action proper to only one or the other is performed, then we EOs say that the action was performed in one nature or the other, as Cyril stated: "Christ suffered according to the nature of humanity".
As a result, in common speech, we can also say as a manner of speaking that the human nature or the humanity suffered, without actually making a "nature" a "person".
Thus the Greek Orthodox Archdiocese of Australia writes:
Jesus wept at the death of Lazarus because He loved him and also because He saw the corruption of human nature after sin and penetration of death. Man was not created to die, but sin introduced mortality. Christ is not only God by nature, but also man. Therefore His human nature suffered.
An example in human nature, is the relationship between our mind, its ideas and the expression of these ideas in words.
Christ is one person, yet has two natures. His Divine nature is united with his human nature - without change, confusion or division. ... "Without division", means that the two natures never separated, so when Christ did what was Divine, His human nature followed, and when the human acted, His Divine nature co-operated. Each nature acted "in communication with the other".
When it uses the expression underlined above, the author does not think that the human nature is a separate "person". You are right that it is Christ who performs the actions , but the natures cooperate and as a manner of speech we can say that they perform actions too.
There are many examples of this in common speech, like saying "a crime against humanity" or "You have insulted my humanity", or:
It is a process, covering the preservation and conservation, economically and beneficially, of everything, edible or potable, that human nature eats and drinks
Pure Products - Volume 14 - Page 471 https://books.google.com/books?id=m2DOAAAAMAAJ
In none of these cases do the speakers think that the natures or the "humanity" refer to separated persons.
This goes back to the issue of flexibility of language and whether we want to understand expressions used in different normal ways as a manner of speech or aim to find problems in order to infer that the other side has mistaken christology.