Same reason he didn't just drop the stadium on the White House. They'd want to search ever crook and cranny at Alcatraz for secret areas and stuff the bridge drop wouldn't affect.
I think, actually, with the stadium, he created a defensive ring for himself around the White House so it would be a lot harder for them to get reinforcements.
Edit to add: And if he dropped the stadium on the White House, he wouldn't have had an opportunity to give his speech to the cameras and the world. He woulda been shouting from the sky as everyone ran away. So, that wouldn't work.
And, also: how do you know what they'd want to do in Alcatraz? They wanted to destroy the cure and the source of the cure. That we know. They didn't talk about, "oh, yeah, and let's find the secret rooms, I bet you there are secret rooms." That's just conjecture on your part.