Veeps keep off the presidential lawn

There seems to be a new paradigm in US politics.  Usually, two-term presidents give way to let their vice-presidents have a crack at running for the office of chief executive.  Sometimes they succeed as George H W Bush did in 1988 and sometimes they fail as with Richard Nixon in 1960 or Al Gore in 2000 (the Supreme Court deciding that latter election).  It usually is a one-term presidency as the public get fed up with one party occupying the Oval Office and the incumbent party gets complacent - witness Bush the Elder looking at his watch during a presidential debate, alienating an evangelical conference by declaring, "Looks like I'm the only person here born once," and slapping aid restrictions on Israel to force them to the peace table with the words, "Jewish widows in Florida don't vote Republican anyway" (ironically in 1992, he won the state of Florida).  Although technically Calvin Coolidge was a one-term presidency, taking over midway through a term after the timely (given how disastrous his rule) death of Warren G Harding, Coolidge's successor, Herbert Hoover took over just as the Great Depression was about to strike and then made about every mistake possible to turn a dire situation even worse.  Franklin Delano ushered out 12 years of Republican rule and instituted 19 years 10 months of Democrat rule (for during FDR's tenure, inauguration shifted from March to January).  Jimmy Carter's running mate on the ticket, Walter Mondale practised a sui generis approach.  Becoming vice-president in 1977, he lost with his boss in 1980 to the victorious Ronald Reagan and so with two elections as a running mate behind him, he stood as the Democratic Party candidate of 1984 - it was a virtual wipeout, Reagan taking 49 out of 50 states.
But things have changed.  Dick Cheney was the first vice-president who serves two terms under a two-term president not to run for the presidency since Thomas R. Marshall, Woodrow Wilson's Veep between 1913 and 1921, so almost 100 years.  In fairness, for a lot of the time, Cheney was the de facto POTUS (President Of The United States) and thus had already exercised supreme power.  Also, his 9% approval ratings (where 91% did not approve of him) threatened a humiliation had he sought the Republican nomination, let alone pitched battle against the Democrats in 2008 (the first time since 1952 in which the election was not contested by a sitting president or vice-president).
Now, Joe Biden, 'the happy warrior', has ruled himself out of running in the 2016 election and, given his age, effectively future presidential contests.  He is still recovering from the devastating loss of his son from brain cancer and this is the official reason why he is not making a run.  He knows better than most how his gaffes could come back to haunt him (after becoming Barack Obama's running mate in 2008, he told a journalist that Hillary Clinton would be a better vice-president than he).  Also, I think he was irked by being the ABC candidate (Anyone But [Hillary] Clinton) rather than being wanted on his own merits and would have hated being the candidate who stopped the most electorally credible woman in US presidential elections from assuming the historical mantle of first female president of the United States of America.
So very different reasons for the two vice-presidents since 2001 to serve as back-up for two terms but not to seek the highest office.  Presidential candidates tend to seek running mates who will not serve as a counterpoint to their authority if they are fairly confident of victory - it will be interesting if the pattern is repeated in 2024 (or 2028 if the next president is only one term).