I don't see how the distinction between those two kinds of properties is relevant. X exists as long as X instantiates properties, whether accidental or essential. X is an object, so we say of objects that they exist.
All we are saying when we say that X exists, is that something Xes. And when we say that X does not exist, we are saying that nothing Xes. So if we say that, for instance, Pegasus exists, what we are saying is that something pegasizes. And Pegaus does not exist means that nothing pegasizes. ( is a winged horse owned by Bellerophon).
I'm not sure how important the distinction is either but I think its worth noting.
For one thing it seems that for something to exist it must at least one essential
property. If X had only accidental properties and no essential ones would it still exist?
X exists if it instantiates at least one essential property. X has at least one essential property if X exists.
This seems to make essence and existence equally important. You can't have one without the other.
Essence doesn't precede existence (like Avicenna said of all things) nor does existence precede essence (like Sartre said of at least some things). They are simultaneous.