Count me another vote for LBJ. If by “politically effective” you mean the ability to carry out a political program, then Johnson shines as does no other President since FDR. There is no other post-WWII President with accomplishments to rival those of Johnson.
However, if maintaining personal popularity is an important part of the analysis, then I look instead to Reagan. He was elected by strong majorities both times, and his designated successor was elected by another convincing majority when Reagan stepped down. In addition, Reagan had one of the most consequential political programs in history, with changes in the philosophy of government that reverberate to this day and the successful management of the Cold War. Whatever you think of their domestic programs - and I personally think they were disastrous - we are lucky that Reagan and GHWB were there at the end of the Soviet Union.
He never struck me as a particularly good politician but Dwight Eisenhower probably did the most to make the USA the supreme military superpower out of the group. His biggest accomplishment went unnoticed by the public as his role in developing the US Spy Satellite program wasn’t declassified until 1992.
Nixon won California by less than 36,000 votes out of more than 6.5 million cast. I’m not alleging any kind of fraud, but there was a patchwork of blue and red areas in the central part of the state which could have gone either way.
I spoke of their prior government job title, not their electoral history. Washington’s prior title was General. Sorry, I erred on the other two; I must fix my presidential database. Before the presidency, Harrison’s prior title was Minister to Gran Columbia. Jackson had indeed last been a US Senator although that wasn’t his idea and he resigned before long.