Hi chestanch,
Welcome to WebProWorld!
You are not alone, that's for sure.

Originally Posted by
chestanch
I am finding more and more clients was to update the sites themselves.
Ofcourse they do. Either because they want more control or because they want to save money... usually the latter
What do you usually recommend to these clients to do for updating their own sites?
I tell these clients that if they update their own sites, they might break the site or the design.
And if that happens, I will have to charge them double to fix it.... - that's usually all I have to say.
If you start offering these "design tools" or a cms program, then it's pretty much a guarantee that you will also be fielding questions and asked to teach them. - definitely not something most web designers are willing to do.
Code:
Does your design firm teach the client how to use a third party software such as Contribute?
Well, I suppose it's an option, if you really want to.