I think the system we have now is pretty good. I just wish it were better enforced.
There are two big aspects of the American approach I find troubling:
- exempting churches from normal reporting requirements
- the clergy housing allowance
Apart from those, my impression is that in the US, something like a non-profit social club would get tax breaks like a full-blown charity (IOW, you don't make tax distinctions between "charities" and "non-profit organizations" like we do up here), so if that's the case, then in general, I'm okay with the current system.
... though I'm sure that there are plenty of "churches" that are really for-profit businesses, but hopefully just making churches follow the normal reporting rules would solve that problem.