The Adamses were indeed highly educated. That didn’t make them successful in politics. Sometimes formal education can get in the way of practical learning.
Not so much “educated” as practical geniuses, and dedicated public servants. John Adams gave up a great deal of personal income to serve as a diplomat in Paris and London, and as a representative in the Continental Congress, and as President. Adams was the most successful lawyer in Massachusetts, and he enjoyed performing manual labor on his small but productive farm well into his sixties.
Adams ran afoul of Alexander Hamilton, who flipped New York against him when he ran for reelection, handing the election to Jefferson. There was a populist movement for war with France over French deprecations against American shipping, and the XYZ affair. The United States began to raise an army, and former President Washington, who was suffering from old age, was appointed to lead it. He selected Hamilton as his aide-de-camp, against Adams wishes. Adams saw Hamilton as a Napoleonic (or Caesar-like) threat to the young republic, but deferred to Washington. Because of Washington’s age, Adams feared he could not reign in Hamilton.
Leading up to the election, Jefferson secretly employed a “journalist” to produce a newspaper that printed false and scurrilous material about Adams. The original Steele Dossier. After the election the connection between Jefferson and the “journalist” came to light, leading to the estrangement between Jefferson and Adams. While Adams, and his wife, considered the office a burden, he was keen to keep the U.S. from war, while Jefferson, like Hamilton, was eager for war with France. Adams considered keeping the U.S. from war with France his greatest achievement. History celebrates those who lead nations into war, but forgets those who averted them. Adams was glad to be forgotten.
Adams got some measure of passive revenge on Jefferson. When Jefferson did not reward the “journalist” who defamed Adams with a diplomatic appointment (he wanted to be ambassador to London or Paris) but merely a local postmaster’s post, he turned his bilious vitriol against Jefferson. He is the only contemporary source for stories about the relationship of Jefferson and Sally Hemmings.
John Quincy Adams was, if anything, even more brilliant than his father. He served with the U.S. Diplomatic mission to St. Petersburg when he was 14, and went on to an outstanding career, fully worthy of his legacy.