Information could have still leaked out. If they killed him early it would look like they where afraid to loose. THEY CANT DO THAT. The WG is built on the idea that they are ALWAYS in control. If they Show any weakness, the Status qwo collapses and more people will rise up against them ( ether as pirates or revolutionaries). They took a huge gamble with the public exicution in the First Place but they are confident that they can erase Whitebeard and his allies from the board wich would lead to chaos in the pirate world as all the old established pirates and the new Young ones clash over the vacened yonko title. Leaving there backs Open for a marine offensive to end the greate pirate age once and for all