Questions:
Have you (they) never had a disaster where you (they) needed to revert to a previous version of the web site but couldn't because they'd broken it?
Do they use a staging web server to test changes?
Surely they don't modify the code in the production server without some testing somewhere?
I suspect the answer to the first is "yes (they have had disasters)" and the second "no", and I hesitate to guess the answer for the third (but. from the sounds of it, the answer might be "yes, they really do"). If that's correct, I admire their bravado and am amazed that they never make a mistake. I'd never risk making changes direct on a live web site.
I would recommend using a version control system or VCS (any VCS) yourself. Work out the wrinkles for the code you look after, and develop the slick distribution that makes it easy (probably still using SFTP) to distribute the VCS code to the web site. But also show that preserving the previous versions has its merits - because you can recover who did what when. To start with, you might find that you need to download the current version of whatever page (file) you need to work on and put that latest version into the VCS before you start modifying the page, because someone else may have modified it since it was last updated in your master repository. You might also want to do a daily 'scrape' of the files to get the current versions - and track changes. You wouldn't have the 'who', nor exactly 'when' (better than to the nearest day), nor the 'why', but you would have the (cumulative) 'what' of the changes.
In answer to comments in the question, Ólafur Waage clarified that they've had disasters because of the lack of a VCS.
That usually makes life much easier. They goofed; they couldn't undo the goof - they probably had annoyed customers, and they should have been incredibly annoyed with themselves. A VCS makes it much easier to recover from such mistakes. Obviously, for any given customized site, you need the (central) backup of the 'correct' or 'official' version of that site available in the VCS. I'd probably go for a single repository for all customers, using a VCS that has good support for branching and merging. That may be harder to deal with at first (while people are getting used to using a VCS), but probably leads to better results in the long term. I'd seriously consider using on of the modern distributed VCS (for example, git), though lots of people use SVN too (even though it is not a distributed VCS).