> the individual states were essentially their own countries with individual governments and leadershipThe debate over the scope of the federal government dates back to before the end of the revolutionary war, and was a central topic of debate in the drafting and ratification process for the Constitution.
Remember that our first attempt at forming a country erred toward a weaker federal government, and was more-or-less an abject failure.
> Where is it stated exactly what the purpose of the electoral college was, in the Founding Fathers' own words?
Federalist 68, in which almost every argument makes literally no sense when compared to direct democracy.
That paper is mostly arguing for the electoral college over e.g. the "Governors" or "congressional" plans, leaving the infeasibility of direct election as a foregone conclusion.
A lot of founding fathers would've preferred a direct vote. See for example Anti Federalist 72. Using google you can also find quotes from Madison, for example, arguing that a direct vote would obviously be best.
Which begs the question: why not just do the obvious thing?
> The only way they could make the deal work with all of them was to ensure that no one state could "overrule" the others
Why don't we come out and be explicit about it -- the only way to make it work was the make sure that slave states were comfortable that they'd be able to retain political power while continuing to subjugate a huge portion of their population.
> If smaller states could be bullied by larger states, the term "united states" loses a quite a bit of its meaning.
Larger by what measure?
The slave states that were opposed to direct election and are today characterized as "rural" weren't actually significantly less populous than the northern states. Virginia -- a slave state -- was the most populous.
It's just that a huge number of their men were black, and so didn't count in a direct election.
So, the whole "rural / less populous states need a voice in presidential elections" thing is complete and utter horse shit. The actual issue was that "very populous states that choose to treat a big portion of the population as sub-humans need a vote controlled by whites but with power proportional to their entire population".