This is the dirty little secret of billionaires' private foundations. That 5% has a lot of latitude and little oversight. For example, it includes administrative costs. A private foundation can pay each family member a salary (allowance) and that's a valid cost against the 5%.
As for the Mormon church (or any church for that matter) non-profits aren't meant to be politically active, as in campaigning for a particular party of candidate. We all know this happens all the time..
I'd be fine with keeping tax-deductability for contributions to complying non-profits. But why do they get to enjoy tax free status on, say, investment gains? Or income from investments (including property they own)? Get rid of that and I think a lot of problems go away.