It looks like the culprit was indeed that plugin TinyMCE Advanced. It is not compatible with Aesop Story Engine.

My Aesop component code was swapped from:
[aesop_gallery id="402"]

… to:
<div class=”mceItem aesop-component-short aesop-component” data-mce-placeholder=”1″ data-aesop-sc=”%5Baesop_gallery%20id%3D%22402%22%5D”>
<div class=”aesop-component-mask mceNonEditable unselectable”> </div>
<div class=”aesop-component-bar”>
<div class=”aesop-component-controls”>
<div class=”aesop-button aesop-button-delete” title=”Delete Component”> </div>
<div class=”aesop-button aesop-button-clone” title=”Clone Component”> </div>
<div class=”aesop-button aesop-button-edit aesop-scope-gallery” title=”Edit Component”> </div>
<div class=”aesop-button aesop-button-clipboard” title=”Cut Component / CTRL + ALT + ENTER to Paste”> </div>
</div>
<span class=”mceNonEditable aesop-component-title unselectable aesop-gallery-title”>gallery</span>

</div>
<div class=”aesop-end”>WcMgcq</div>
</div>

I removed the plugin and the problem is now resolved.