There's also a philosophical reason for the adoption of an indirect, representative democracy. One specific example is to protect against Tyranny of the Majority [1].
I'm dubious that a country where most issues are voted on directly by the people could function with any significant amount of participation, mostly because people are not experts in most fields, and don't necessarily understand e.g. the macroeconomic effects of letting the largest banks in the world all go bankrupt. We elect representatives because there is a lot of work to do governing a country thousands of miles wide with over 300 million people, and I want people who can devote their profession to improving government in charge. Ones that we get to vote for regularly, of course.
"Second. It is of great importance in a republic not only to guard the society against the oppression of its rulers, but to guard one part of the society against the injustice of the other part. Different interests necessarily exist in different classes of citizens. If a majority be united by a common interest, the rights of the minority will be insecure. "
The context of your Federalist Papers quote is in regards to separation of powers (legislative, executive, judicial) and not indirect vs. direct democracy.
Tyranny of the majority and mob rule are just pejorative synonyms for democracy. Which would you rather have, tyranny by the majority or tyranny by a minority (i.e. monarchy/oligarchy/dictatorship)? Personally I'd argue for something else altogether.
At one end of the debate was the likes of Thomas Jefferson and John Adams who argued for the "tyranny of the majority" (democracy) and at the other was the likes of Alexander Hamilton, who wanted Oligarchy:
>I believe the British government forms the best model the world ever produced, and such has been its progress in the minds of the many, that this truth gradually gains ground. This government has for its object public strength and individual security. It is said with us to be unattainable. All communities divide themselves into the few and the many. The first are the rich and well born, the other the mass of the people. The voice of the people has been said to be the voice of God; and however generally this maxim has been quoted and believed, it is not true in fact. The people are turbulent and changing; they seldom judge or determine right. Give therefore to the first class a distinct, permanent share in the government. They will check the unsteadiness of the second, and as they cannot receive any advantage by a change, they therefore will ever maintain good government.
-Alexander Hamilton, Farrand's Records of the Federal Convention, v. 1, p. 299.1787-06-19
It would indeed be nice to delegate to the experts (i.e. scientists, engineers, psychologists etc.) on relevant issues, rather than having climate change deniers[1] on the Environmental Committee and people who have never sent an email[2] in their life on the Committee for Privacy and Technology.
I'm dubious that a country where most issues are voted on directly by the people could function with any significant amount of participation, mostly because people are not experts in most fields, and don't necessarily understand e.g. the macroeconomic effects of letting the largest banks in the world all go bankrupt. We elect representatives because there is a lot of work to do governing a country thousands of miles wide with over 300 million people, and I want people who can devote their profession to improving government in charge. Ones that we get to vote for regularly, of course.
[1] https://en.wikipedia.org/wiki/Tyranny_of_the_majority
"Second. It is of great importance in a republic not only to guard the society against the oppression of its rulers, but to guard one part of the society against the injustice of the other part. Different interests necessarily exist in different classes of citizens. If a majority be united by a common interest, the rights of the minority will be insecure. "
- James Madison, Federalist #51 : http://www.constitution.org/fed/federa51.htm