Fix SSE edit_area sync bug

What does this MR do?

Fixes a bug that otherwise let the static site editor work as a side effect of the mode change (from WYSIWYG <-> Markdown). Previously if a mode change did not occur, then the sync would not occur properly as expected. Now the sync works as expected regardless of the lack of a mode change.

Screenshots

N/A

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Closes #221289 (closed)