I said "grammar" but you are right, that isn't actually what I meant. It's syntax and pronunciation and in some cases, making up words, as in "they misunderestimated me."I know you are a professional writer/communicator so I hesitate to contradict you on language issues. However, my own observations cause me to believe the opposite regarding presidential grammar.
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President Bush used to at times, butcher syntax and sometime the pronunciation of certain words (Nuclear?). Notably however, he assumed his listeners had more than minimal intelligence and accordingly never spoke down to anyone or dumbed down his comments.
President Obama seems to be preachy, and commonly speaks as if his listeners are stupid. Additionally, he routinely drops his "g's" and uses slang and colloquial terms, even when talking to sophisticated audiences. (" Ya don't go messing with" ... stupid "Stuff" etc. etc.) It often seems that he just does not know of words or phrases that would be more suitable. His rambling conversational answers often leave him grasping for words and he comes up with street slang.
Overall, I never found myself going "Huh? What did that man just say?" after listening to President Bush ... something that happens time and time again with President Obama.
President Bush was notably concise and often this led the easily-fooled to be fooled. Some people think that the more words you use, the more articulate you are. President Obama benefits from this misperception.
You also are right about Obama. He uses street slang because he doesn't think his listenders, generally speaking, are smart enough to understand him if he talks like the highly educated person that he is.
But he's entirely different when he's faced with people who disagree with him. I think the best description of his speaking style in those cases that I've ever heard was that he sounds like a college freshmen, home for Thanksgiving, lecturing his parents on the way life is.
Can't leave this topic without mentioning Jimmy Carter. One of the almost surrealistic things about him was his mastery of English usage. You could take a transcript of his extemporaneous remarks and diagram the sentences, with every part of speech in the proper place.