Framework
While it's a good idea to use a Joomla website, it's probably not a good idea to have such an old installation sitting around waiting to be hacked. In fact, I don't even remember when the login page looked like the one (that is easily accessible) at Cross Connections. The newest it could be is 1.5, and we've since upgraded to 1.6, 1.7, 2.5, 3.0, 3.1, 3.2, 3.3, 3.4, and 3.5.
Mobile-Friendly
Not so much. Here are the problems that Google sees:
- Content wider than screen
- Clickable elements too close together
- Text too small to read
- Viewport not set
Other Issues
I can scrape about a dozen emails right off the website, which means mass spam is sent to those who have their addresses on the site. It's a big website with a lot going on, so it deserves a new Joomla version. However, with its age, you might consider a rebuild on a new Joomla framework, since there are two big migrations to be made to get it up to current Joomla. Since it's on GoDaddy, it can be done, but I know there are lots of forum posts about how it's not always so easy. Website owners could pay $2000+ for upgrade of this scale. Luthernet would probably be about $1500 to rebuild and host it for $100 a year.
A Simplified Guide to Church Websites: Purpose, Planning, and Presentation