Yes, he will cool off and get back to you.
He will most likely do it sooner, as long as you refrain from sending him anymore messages.
If you keep giving in to him by reaching out because you are fearful that he will stay mad for too long, he will keep acting up every time he doesn’t get his way.
On the contrary, if you let him start contacting you first, it will only be a matter of time before he stops this kind of behavior because it will simply become too emotionally exhausting to keep doing it when he knows that he won’t win.
As you stated, you have repeatedly failed to let him contact you first. Hence, you know what you need to work on.