We weld all the differental gears on our race cars. The positives are so much greater than running an open differential to the negatives. For starters, between two cars, we'd have to run five differentials with five different LSD units. That would equate to about $7k. That's bonus enough right there for us.
As far as the downfalls, it'll make it harder to push, that's for sure & take a little more juice to get around a slow corner. And, yes, the turning radius will decrease but golf carts turn so well to begin with it's not going to be noticeable. This is all negligeable compared to the overall traction you will gain.
I'm not sure what stripping out hubs has to do with the welded diff. For some reason I just can't associate the two.
The trick to doing this is, if in anyway you can have the axles in the carrier bearings while welding to help take up any slack. The other necessary thing is to basically spot weld it a little at a time and build the weld up as you rotate the gear assembly. That way the diff gears won't get too hot and carry the heat to the bearings. Other than that, clean out the slag real well, put it back to gether and romp the hell out of it.
Please keep in mind, I've never even taken apart the rear diff on a golf cart so it may be compeletely different than a regular automotive type application.
If you'd like, I can snap a photo of one or two of our differentials so you can see what I'm talkiing about if you're unfamiliar.