HTML API: Remove unused processor state context_node property

[sirreal]

The processor state context_node property is unused and redundant. Remove it.
The processor uses a context_node property on itself.

The $state property is private, so it should not be accessible to subclasses.
The class itself is marked with access private so should also be unused.

See #7348 (comment).

Trac ticket:

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[github-actions]

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