currently the burnside special, both pessimas, the covet and possibly other cars don't have the option for a welded differential, which severely limits their ability to drive while damaged, and I don't really see any reason that they wouldn't be able to have a welded diff although I'm not really a car expert.
The main reason people use welded diffs is for drifting AFAIK. Most of the cars you mentioned are FF, so are not likely to be used for drifting anytime soon, and the Burnside is too old and valuable to do that with.
I know that's why its normally done but that doesn't mean it doesn't have other uses, like if we can replace single parts instead of fixing the whole car in career mode it could be a cheap alternative to fixing a bent chassis, or a missing wheel Edit: I'd also argue that some people would be dumb enough to weld the diff on a front wheel drive car to try to get it to drift...