1) Excelling in the election process is no indicator that you have the necessary skills to be a good president. In an increasingly polarized country, elections are won by refusing to compromise with the other side. This can lead to situations akin to the government shutdown over Obamacare, which is not reflective of a successful government. Eliminating presidential elections (and elections at all levels) would allow for a system to be put in place which would promote leaders, not just people who can get the votes.
2) The system is biased. The two major political parties make it very difficult for third party candidates to even run for elections. Oklahoma doesn't even list third party candidates, such is the dominance of the republican and democrat parties. This severely hinders the ability to choose the best candidate, as he must be affiliated with the republicans/democrats, and must therefore adopt the party views. This limits any president who is elected to acting within their party's ideological boundaries, and prevents a president who may compromise between the two from being elected.
3) The people can be wrong. There is an implicit assumption in the election process that the people can choose the best leader for the country. Jon Stewart best explains this: "they don't go with the best, they go with whoever won."; surely the best leader is the most desirable? Throughout history, there is a lot of evidence that the people cannot choose the best leader, only the most desirable (an example being Hitler, whose leadership ultimately did not benefit Germany).
A democratic election system is not the only system of government, and there is a lot of evidence today that it is time to try a different government, as the inter-party (and intra-party) bickering has reached a point where it is impossible to move the country forward.
Without elections, how would our leaders be determined? There would have to be a committee of leaders, probably not elected, to determine our leaders. This is NOT a good idea. Voting is a way for people to show their support for not just a candidate and a party, but also the ideas represented by that party and politician.