The DOM is a model of a markup language instance plus access methods. State describes memory of properties resulting from a change. For example HTTP is stateless.
The DOM can be both stateful and stateless, neither, or one of. The statefulness depends on your application logic rather than any intrinsic quality of the DOM itself.
Any stateful data structure is stateless if it’s not changed. That doesn’t seem to me to be a particularly deep revelation about the DOM.
Back to your original point, that code example you linked attatches a shadow dom, creates an element in it and then puts text in it. I fail to see how that’s declarative or stateless, and AFAIK that’s the only way to populate a shadow DOM (the original HTML of the document cannot put elements there).
The DOM can be both stateful and stateless, neither, or one of. The statefulness depends on your application logic rather than any intrinsic quality of the DOM itself.