Why SHOULD the Speaker of the House be second in line to succeed to the office of the President?
In my opinion, the main reason for this is that the House of Representatives is the "people's house." It is the part of government that is closest to the people. This was especially true back in the days when Senators were not elected by the people.
Clearly, the person who should be first in line is the vice president. But after that person, who? It makes sense to me that it should be the leader of the House. The members of the House are elected more often than Senators. They are supposed to be the members of government most in tune with the opinions of the people. By having the leader of this house be third in line, we send the message that the will of the people is what is most important in our system of government.