to chant a funeral dirge: to perform the composed song of grief over the dead, or over a nation mourned as though it were already dead
This verb is not a word for crying. It is a word for performing. It means taking up the composed funeral song and chanting it aloud over the dead. The song itself has its own name, the dirge (קִינָה), and this verb is what a mourner does with one. When David chanted over Saul and Jonathan, the text quotes the whole poem and adds that he had it taught to the people of Judah, which tells you it was a finished piece of work and not a burst of feeling.
Every use of the word sits in public. Someone leads and others follow: the king chants over Abner and the whole army weeps with him, Jeremiah chants over Josiah and the singers keep his laments as a standing custom for generations, and the mourning women are sent for precisely because chanting laments is their trade. Grief here is something a community performs together, with skilled voices at the front of it.
Ezekiel does something startling with the word. He chants the funeral of Tyre and of Egypt while both are still standing, so the song turns into an announcement: the fall is so certain that the mourning can begin already.
It helps to keep the mourning words apart from one another: