Error 404 Component Not Found when enabling SEF URLs

What?
So I'm getting an error like this:

The requested page can't be found.
An error has occurred while processing your request.

You may not be able to visit this page because of:
- an out-of-date bookmark/favourite
- a mistyped address
- a search engine that has an out-of-date listing for this site
- you have no access to this page

If difficulties persist, please contact the System Administrator of this site and report the error below.
404 Component not found.

Just because I'm doing something like this:
How?
I found that this was only affecting my subdomains where this template was installed. I installed some terrible 3rd-party SEF plugins to correct the situation but when I moved my test site to live at www.joellipman.com (instead of being on http://test1.joellipman.com), I found I could disable the third-party SEF extension and enable my Joomla SEF settings... ??? This is a Joomla 3.x issue.