Adams had a reputation amongst his opponents as divisive and stubborn. However, that stubbornness emerged from a sense of deep commitment to republican values and principles. That commitment served him well throughout the 1760s through the end of the 1780s. He had a great deal of respect, even from many of his opponents, and was an intellectual powerhouse who wielded enormous power in New England. His reputation would only significantly fall during and after his presidency.

Adams, by 1788, was very much amongst the first tier of American statesmen. He had pushed stronger, earlier for independence than virtually any other senior colonial politician. His work in the Continental Congress was vital to the establishment and survival of the United States. Furthermore, while certainly egotistical, he had enough self awareness and perception to work behind the scenes on divisive topics and allow more moderate, congenial voices to be the face of those topics. There's a reason Jefferson wrote the Declaration, and it wasn't because Adams lacked the eloquence or capacity. Adams also had an acute awareness of the sectional divisions between the colonies / states. He was as staunch an abolitionist as you would find amongst the upper classes in the late eighteenth century, but he moderated his moral principles when it came to the unity of the nation. Adams helped nominate Washington as commander of the Continental Army because he wanted a Virginian to lead the New England militias around Boston. It projected a unity that did not necessarily exist in 1775. He led many of the debates over revisions to the Declaration of Independence, making key concessions that satisfied delegates from the lower South.

Adams was an astute politician and political theorist. His 1776 tract, Thoughts on Government, served as a model for many of the new constitutions states produced on the verge of -- or shortly after -- independence. Wood calls it, "the most influential work guiding the framers of the new republics," (The Creation of the American Republic, 568). Adams recommended a government with executive, legislative, and judiciary branches that checked one another. He also recommended avoiding a single legislative body; he worried that such a government would devolve into oligarchy or cronyism. He was able to put his ideas into practice in 1780. The Massachusetts Constitution that he helped lead the drafting of is still in effect today and the language in it was responsible for ending slavery in Massachusetts by 1783.

Adams' work overseas during and after the Revolution did several things for his political stature. For one, it meant he avoided the most contentious domestic debates. He still regularly corresponded with allies on domestic matters, but did not have to get his hands dirty. John Hancock, another Massachusetts native with strong political ambitions (and a better personality for politics to boot) lost support with the elites in Massachusetts during his time as governor, which harmed his national chances. It should be said, Hancock remained extremely popular with the common voters of Massachusetts, essentially becoming governor for life. Overseas diplomacy also gave Adams valuable insight into European politics that Washington appreciated during his terms in office.

When it came time for the presidential election in 1788, Washington as President was a foregone conclusion. However, this was not a one horse race. 10 candidates announced their intentions to run for office. For anyone reading this who may not know, there were no parties and there were no presidential tickets in 1788. Voters chose electors who voted for the President. The person who finished second became Vice President. Voting itself was seriously limited. Something like 2% of the population voted in the election of 1788 and 5 of the 13 states had no statewide vote; the state legislatures chose the electors and in the case of New York, the legislature deadlocked and no electors were chosen.

This meant that the choice was left mostly to the political elites of the states. From the beginning, people like Jefferson and Madison expected the election of Washington as President to be sectionally balanced with the choice of a New Englander as Vice President. Adams was still the dominant political force in New England and could count on their support. To use the examples you provided to show why Adams received more support: Jefferson was another Virginian, Hamilton wasn't 35 in 1788 (and not a national figure yet), and Clinton was an anti-federalist who could not expect support from federalist state legislatures. Adams made sense as a choice.

Once in office, Adams strongly supported federalist policies and cultivated allies in the growing partisan environment. And, while he was obnoxious at times, he was a man of honor and honesty. He largely avoided the dirty politics of Hamilton and Jefferson while maintaining the support of New England as a block. It helped smooth the way for him to assume Washington's mantle.