You half halt with your body. It is really tough to explain the half halt over the internet, but basically you are rebalancing the horse, asking it to come onto it's haunches while becoming more active in the hindlegs. Because pulling on the reins nearly immediatly halts any hind end activity, by doing that during a half halt you are basically screwing yourself. If I ever get a video of me riding I'll find a half half and post it.
Yes, in fact there is a controversy over that whole thing in the FEI right now. Double Bridles are currently mandatory at most FEI levels (Juniors, Young Riders, PSG, I1, I2 and GP) and this restricts some horses from competition who can't handle a double bridle. Because we are breeding horses more sensitive, many of them don't need the double, or go completely screwy with it on.
Still, even in the snaffle, the rider should not have to drastically change position to get a different aid.