Of course Washington was a political creature motivated by a desire to achieve fame, power and position. However, his ambition was also to be seen as worthy of that power and position because of his character. In the 18th Century character was seen as not only what you were but what you aspired to be and what you were seen to do. Therefore, a man of character was a man of personal and public honor and a man who put the needs of his nation above his own desires. Washington was such a man and we were damn lucky to have him.
Washington was not dieing or sick when he left the Presidency. He died suddenly of what was probably strep throat and bad doctoring. He was, however, deathly sick of politics and scandal mongering and home sick for his farms. He had no further ambition for power and glory and was happy to leave. Just as his every other act while President created a precedent for his successors, his leaving created a precedent, and he knew it.
Washington was, indeed, formal and distant in his conduct of the office of President. Partly this was due to his nature. He was not normally a boisterous fellow, partly because he believed the new, rough nation needed a bit of formality and dignity. I do not believe he was mocking George III, he was trying to create a nation out of some very rough materials.
Although the concept of an American monarchy, elective or otherwise, is a fascinating topic for AH, I believe that without a major POD which totally changes the nature of the American Revoultion it will remain in the realm of highly unlikely to totally implausible because Washington was the only one who could have started it and he would have nothing to do it with it.