MariaDB Multiline RegEx
Just through trial and error and persistence, I found a solution after about 2.5 hours:  

select REGEXP_REPLACE(post_content,'(*CRLF)&lt;noindex&gt;.*&lt;/noindex&gt;','') from wp_posts where ID = 318


If anyone could explain why the above solutions in my question didn't work though, that would still be appreciated. 

Reference to (*CRLF) on this page: 
https://mariadb.com/kb/en/library/pcre/#multiline-matching