As a gut feeling, the proportion of MPs who want to be Prime Minister would be at least equal the number of Congressmembers who want to be President.
Now, sure. But at the time of the American Revolution, was there even a British Prime Minister, and did they have any significant level of power?
Yes, of course. There had been very powerful PMs like Walpole and Pitt the Elder, middling ones like Bute. During the time of the Revolution, the main PM was Lord North (a middling one).
The king definitely had more power in the 18th century than later in the 19th, but Cabinet government steadily evolved throughout the 1700s, weakening the King’s power. By George’s time, the King had to have support from Parliament.
That’s why the Declaration of Independence is politically deceptive, in putting it all on the King. There is no way that the King could have done all that was done by the British government in the lead up to the Revolution, all on his own. He had to have parliamentary support.
For example, the Intolerable Acts - those were passed by Parliament. They showed that the upper levels of government, Parliament and King, were generally agreed in how to respond to the colonists. But, there were opposition members in Parliament who opposed the harsh approach to the colonies.
But, it was politically easier for the drafters of the DoI to put it all on a tyrannical king, rather than to admit that a legislative body that had, for the time, a strong popularly elected component, had agreed with the measures leading up to the war.
And at the time of the writing of the Constitution, the PM had been for several years William Pitt the Younger, who was a commoner. And not the first. The Founders would have very conscious that the position was coveted by many and available to all the elite, not just the nobility.
George III was violently opposed to American independence, and apparently took the DoI as a personal slight against him, good propaganda that it was. Though Parliament controlled the country, the King was vastly more influential then than today. George’s intransigent stance weakened the power of those against the war, even Lord North who realized early on that the war was unwinnable. George kept North on, regardless, because he seemed the best to abide by his wishes. North was therefore forced to resign after Yorktown. Another clear signal that after Washington any President would be vulnerable to political enemies and shifting fortunes, which encouraged his party in Congress to protect and empower him whenever possible.
“Commoner” here in the English sense of “younger son of an Earl”, of course. 
Ooh, a nepo baby.
Of course, his father was known as “The Great Commoner” because he wouldn’t accept a title before he became PM. And he had been Leader of the House of Commons earlier, just as North and Pitt the Younger would be.
I got your joke, although the nuances of the English class system have always been a mystery to me. But politics is politics, with titles or without, and the politics of the Founders was politics.
Here’s a couple of Wikipedia articles about the two main groups during the revolution: the Tories led by Lord North, and the Rockingham Whigs, led the Marquess of Rockingham. Both were prime minister at different times.