No, the FAA's flight restrictions could not do more to prevent another drone flying onto White House property, because there is only so much that the FAA can do. The FAA can't control the solo person down the street tinkering with a drone. They can't shoot the drone down if it approaches the White House. There's only so much the FAA can do.
I think this sort of is irony working out perfectly. As citizens its ok the amount of monitoring our government does on us on a daily basis, however we are all supposed to be concerned about a drone accidentally crashing on the White House lawn. At some point things like this are going to happen.
The FAA can only enforce regulations to the best of their ability. They have no power over the radar's ability to detect small flying objects, especially at low altitude. Creating more regulations and adding more restrictions would have no effect on a person's ability (or willingness) to fly a small craft so close to the White House or any other pertinent, governmental facility. The secret service and military are responsible for their protection against these objects.
No, FAA restrictions wont prevent drones from flying onto the White House property. Restrictions are only words, they do not prevent people's actions. They may deter some people, but ultimately it is a person's choice whether they want to abide by a restriction or not. If someone wants badly enough to fly a drone onto the White House property, they will do so whether there are restrictions or not.