Believe me, intelligence is not by any means a factor in becoming President. Here's a few examples just from our current President. Since I realize you're not American, I'll explain the mistakes.
Obama Claims He's Visited 57 States
We only have 50 states.
Obama: Constitution written "20 centuries" ago
Our constitution was written 2 centuries ago, not 20.
Obama Recalls the "bomb" that fell on Pearl Harbor
We dropped the bombs on Hiroshima and Nagasaki. Pearl Harbor is part of our own country.
I could drop them all for George W. Bush but I'd be shocked if you haven't seen more than enough by now to last you.
Guess if you're not allowed to make a few errors in a speech after having done so much work, no one is fit to be President.
I think the gist is that the race has gotten so ugly in the primaries that whoever finally gets the brass ring won't want to make that offer.
Random thought - in the earliest elections, the Vice President was chosen as the person who got the second-most number of electoral votes. This was because each member had been given two votes, one of which had to be given to someone outside their state. In order to make sure that those 'second votes' were carefully considered, there was an actual 'benefit' to coming in second. After the deadlock between Aaron Burr and Thomas Jefferson, the Twelfth Amendment was put in to ensure that the two offices were elected on separate ballots.
Technically, I could go in and vote for a Democrat President, and a Republican Veep. If it weren't for the Electoral College voting for them as a unit (which is not mandated anywhere in the 12th), it would be theoretically possible to have a bipartisan Executive Branch. Wouldn't that be an interesting wrinkle to throw at a system currently split by party lines?
The country is unmanageable enough with a democratic president and a republican house.
As for a VP candidate, how about Donald Trump? That way the Republicans could tell people they're going to end the crisis with not one but two of the major instigators of this crisis.