The ability to speak in public, particularly on one's feet, has always been considered a prerequisite for political life; Rhetoric was taught as a governing art since way before Claudius' time. Today it is called "communications". Claudius' speech defect was compensated somewhat by his being a writer (and deliberately cultivated to keep him out of the public eye).

I cannot read Mr Bush's mind, but my only glimpse into it through his unscripted utterances shows not only a man uncomfortable with abstract thought, but also one who distrusts it. This is not a political prejudice on my part, Mr Cheney and Mr Rumsfeld, who I find personally much more distasteful than Mr Bush, strike me as being highly intelligent and logical individuals. Bush Sr, although not a skilled orator and a simple, unadorned speaker also spoke in a clear and logical fashion, as did Richard Nixon, a man who rejected "flowery speech" but nevertheless could express himself clearly.

Ronald Reagan was a trained actor, so the jury is still out on him, while the Democrats that come to mind, Kennedy and Clinton, were not only good speakers, but could also hold their own with a roomful of reporters.
Listen sometime to Tony Blair defending himself in a hostile Parliament and ask yourself how long GW Bush could last in that environment.