My experience is ISP and FTSE100 companies and I'd never have even suggested installing a GUI on any of the servers.
My background is ISP and education, with Linux in both, and you're talking radically different worlds. For an ISP sector I'd agree, if you're not capable of handling the command line I don't want you near my servers, and even then it'll be grudgingly, but you do have to be aware that is an ideal world situation, one where no one should be employed or have a need to work with servers that are mission critical.
Education is a lot different. For starters Moodle requires a bog standard LAMP set-up, it really takes little by way of customisation, certainly should be nothing that can't be handled through the GUI interfaces. Secondly it's not a massive resource hog. We had a P4 2ghz server handling our moodle instance with about 1300 students using it, and never once even made the server bat an eyelid, and yes that was with a GUI running on an RHES server. I even had the inbuilt firewall running because I didn't know better. Education also has the advantage that if a server dies and something is unavailable, a server has died and something is unavailable. That's it. You're not costing the company money with every second, you don't have customers businesses that are impacted, you don't have anything like the stress that you get if, say, a RADIUS server dies

It results in a lot more flexibility and a much nicer working environment!
Finally, Public sector doesn't tend to employ people with the skills, and rarely provides the same level of training and opportunities that private sector provides. I'd had Linux running on my box at home for a couple of years (Gentoo, stage 1 build), had been running it on some servers and a media workstation (DVB recording), and had been active on some linux forums, but it wasn't until I started in the ISP that I really had any proper grasp of server control, set-up, best practices and administration.
If you can get away from an elitist, ideal world view of things, what is so wrong with grabbing a copy of a mainstream distribution, and installing a LAMP solution with a GUI? Slightly less resources? So what, it's not like it's going to be hammered hard. Less optimal setup, like the GUI tools usually result in, again not really a big problem.
One of the main distributions is ideal for this work.
However I wouldn't advise Fedora. It's
way too bleeding edge. Great for workstations where the user is convinced they need the latest greatest technology, absolutely awful for servers unless you know to an absolute certainty that you need specific versions of a package that are on it.
Stick with a mainstream server distribution.
From the moodle site, the requirements are
- PHP4 (version 4.3.0 or later) or PHP5 (version 5.1.0 or later)
- MySQL (version 4.1.12 or later), PostgreSQL (8.0 or later) or Microsoft SQL Server 2005
That can even be handled by Debian if I'm not mistaken and that's now getting a bit long in the tooth in comparison to other server distros, but has the advantage of being very stable and reliable.
There is even a moodle package in the apt repository.
If you pick Ubuntu and do a LAMP installation with that, bring up the package manager and you'll be able to install Moodle nice and easily, as someone has put a copy of 1.6 into the repository, unless you specifically need / want 1.9 in which case you'll have to download and set it up yourself which shouldn't be hard anyway.