The {article} tags are not being converted

The Articles Anywhere plugin is not published

Check if the Articles Anywhere plugin is published. Go to Extensions >> Plugin manager and search for the system plugin System - Regular Labs - Articles Anywhere. Then publish it.

There is markup on the {article} tags

If you have copy/pasted code directly into your editor, you might also have copied along hidden HTML tags. So this will also cause the {article} tags to break:

{<span>article</span> ...}

Remove the extra styling or other html tags around and in the {article} tags that might cause problems.

The core "Article" editor button gets disabled upon installation

Correct. Joomla has its own Article editor button. Seeing Articles Anywhere has similar - but much better - functionality, Articles Anywhere will disable the Joomla core editor button plugin upon installation / update.

If you want Articles Anywhere to leave the Joomla core editor button plugin alone, just add articlesanywhere_ignore anywhere in the custom_data field of the core Article editor button plugin in the database.
This requires you to do this manually in the database. You can find the Joomla core Article editor button in the #__extensions table. The row you want to edit will have plg_editors-xtd_article in the name column.