Are you sure you didn't mean CG-spaces (without the Hausdorff assumption)? CGH isn't even closed under colimits (it's pretty easy to construct the real line with two origins), so they don't seem to have all of the nice categorical properties we would want.
I think that the real question is why we need weak Hausdorffness instead of just compact generation.
The above is incorrect (although it seems that Dan Ramras and I have all made the same mistake, as well as the paper cited by Andrey), but here's a paper by Neil Strickland that you might find interesting: Click!.
