A good theory. But would that strategy fly with the public knowing that Obama is constitutionally ineligible for the office? I guess it all depends on the obfuscatory power of the drive-bys.
And what would 'the public' do about it? There is no mechanism for recalling a President beyond impeachment. A President could walk out onto the White House lawn, pull a gun and off a Girl Scout, and there is nothing 'the public' could legally do. Unless Congress is willing to impeach that's all she wrote.