Presidents have rarely been close to their vice presidents or enthusiastic supporters of their successors as party nominee. The vice president or the next nominee were often somebody the president had to beat to get the nomination, so there wasn't much affection or trust there. Or the veep could have been somebody chosen because they were too weak to be any kind of challenge to the president. Since the president and vice president were usually chosen to represent different parts of the party and the country, they may have been too different to get along. Plus, vice presidents were generally kept out of the way, out of the loop, and far from any real power, which would only increase the president's contempt for them. And of course, every president fears that a successor might outshine him. One reason why the Bushes got along better with the Clintons and the Obamas than with Republican candidates could be that they feared being put in the shade and losing their position in the party.
Even when presidents have worked closely with their vice presidents or successors as party nominee and the two have liked each other there are always strains. The president fears being overshadowed by the number 2. Andrew Jackson relied on Martin Van Buren, but one wonders if the war hero didn't come to resent his dependence on the political operator. Carter and Mondale worked closely together, but I suspect Jimmy, the insecure Washington outsider, may have resented the well-connected Washington insider Walter. And of course, Bush and Cheney were friends who worked closely together, but as Bush became more unpopular, he may have started to blame Cheney (as so many other people did).
So sure, for Obama, it's all about Obama, but that's what politics is at the presidential level. When thousands are cheering you and there are signs with your name on them all over the country it basically is all about you, and it's hard to let go of that feeling when it's no longer all just about you. One president who got along spectacularly well with his successor was Theodore Roosevelt, who handpicked William Howard Taft to succeed him. How did that work out?
You seem to distill VP pics to:
1. electoral advantage (regional appeal, etc.)
2. political advantage (internal party politics)
3. personal commonalities (personal/ political)
4. policy commonalities
Trump chose Pence for no. 3, as did TR for Taft. FDR chose Wallace for no.’s 1, 2 & 4, which makes one think that no. 3 was in play as well.
These choices always backfire, the most extreme case likely being that of Lincoln for Johnson. I doubt Lincoln would have lost the election w/o Johnson, but who am I to doubt Lincoln?
As for Taft, idiotic historians have added a no. 5, the relationship between the wives. So pathetic, but they have actually blamed the TR/Taft split on it. The TR/Taft split resulted from TR’s confusion of no. 4 with no. 3. TR himself didn’t understand that his politics had gotten the better of his personality, so that split was inevitable.
As you point out on Bush II and Cheney, no. 4 creates the strongest bond, which I see operating between Trump and Pence. (Run Ike and Nixon through this paradigm, great fun!)
Biden hadn’t a clue that Obama didn’t give a damn for him outside of the (ludicrous) claim that Biden brought the ticket “foreign policy gravitas” (OMG), which falls under no. 1. I suppose there was an internal, no. 2, party rationale for Obama to choose Biden, but the larger purpose was to give cover to the Obama deal w/ the Clintons that the Hildebeast would be SecState then the party candidate to follow Obama.
Easy to see, then, that Obama had no more loyalty or enthusiasm for Hitlery than one might have for riding out a car lease.