So part of what I want to get at here isn't so much, why did it take 10+ years for George Washington to be president, but rather, why George Washington being president in 1789 versus 1784 would be vastly different concepts. Comparing the two positions is comparing apples to oranges. Meanwhile, it is no coincidence that George Washington was the first president under the new government, and under the previous one, it would have been less significant.

Under the Articles of Confederation, there wasn't really a "President" as we know it today. There was no executive branch at all, actually: the states were all loosely affiliated and largely independent of each other, and the only centralized form of government was the Continental Congress, formed during the Revolutionary War. Congress's duties included handling foreign/military affairs, and frankly very little else; each state sent 2-7 representatives to congress, but all of them voted together in a single delegation. (Fun fact: Article 11 says that Canada—and only Canada—is immediately allowed to join the country if it so wishes, but any other land will have to be approved by Congress. I've always found it hilarious how this got outlined in the original constitution.) There was a president of the Congress, whose job was to chair or moderate meetings. It was an important position that held power, but it wasn't that much power, compared to the President under the Constitution. It pretty much just means running meetings.

As you might be aware, the US Constitution was proposed because the loose alliance of states that the Articles of Confederation established was insufficient for meeting the needs of the country, so Alexander Hamilton organized the Annapolis Convention in 1786 to propose the states convene and tweak them a little bit. They reconvened in 1787 for the Constitutional Convention, where everyone knew "tweak them a little bit" meant rewriting it all and starting from scratch. The new Constitution provided more interconnectedness between the states, so that the alliance isn't as loose as it was before, and created a stronger federal government: Congress was more comprehensive, and had a lot more responsibilities; there was a Supreme Court as well as a series of lower federal court, to handle breakings of the law and violations of the Constitution; and of course the executive branch, who enforces the law, led by the President. The federal government was intended to play a much smaller role in the daily lives of citizens than it does today, and the President was supposed to have less power than Congress, but still at the time it was a hefty job that needed a strong leader.

Article II of the Constitution lays out the duties of the executive branch, and in turn the President: leading the military, creating agencies to enforce laws, nominating people to serve in courts and as ambassadors and work in agencies, and so on. It's a big responsibility, one made only bigger as the country grew. And as you can tell, it's a lot more involved than being "president" under the Articles of Confederation, which was essentially just moderating the Congress; now, you had to run an entirely separate branch, in a much more centralized and powerful government. You had to know who the best people to appoint to positions, supervise and work with people on a variety of different levels on a whole bunch of different issues, get Congress to approve of the things you want to do, and be commander-in-chief of the entire military.

So what's the deal with George Washington? Why was he the first president? The simple answer is that everyone loved him. He was a great military leader, a person who everyone respected, and of course, he won the Revolutionary War. A pretty decent résumé. But it goes deeper than that, into actual discussions during the Constitutional Convention. Washington was in fact the chair of the overall convention, meaning he was yet again demonstrating his political insights and abilities and influences. This does not mean he coaxed the Convention into making him President. Rather, they coaxed him; it wouldn't be unreasonable to argue that the Constitution would not have passed had they not believed Washington would be president. This lies in the committee dealing with the Electoral College.

I've written on here before about the complex series of compromises creating the college (an answer that was unfortunately poorly edited but out of principle I will not revise), so I won't go into a whole lot of detail here, especially since most of it is irrelevant. The key thing to note is that the members of the committee had a lot of trouble figuring out how to choose the president, and when the electoral college was proposed, there were a lot of reservations; people were okay with it, but rarely excited by it. It became clear that Washington was almost definitely going to be the winner of the Presidency under this system, and that gave people confidence that the country would work at least in the beginning, so they approved it, passed it on to the full convention, and the rest is of course history. People knew he was a capable leader who would be a good president, and that faith allowed them to pass the US Constitution.

I don't know a whole lot about George Washington's life, so I can't speak too much on what he was doing in the years between the Revolution and the Convention, and why he didn't do anything in federal government. From what I've gathered, he just did not like politics that much and didn't want to be bothered, and then people bothered him into doing it. But even if he were to be president pre-Convention, it would have been a very different job.

Further Reading

Electoral College Primer, Chapter 2 by Lawrence Longley and Neil Peirce (1996)

Alexander Hamilton, by Ron Chernow (2004)

The Articles of Confederation (1777)

The US Constitution, 1787

The Federalist Papers, particularly #67-77 (1788).