The simplest way to answer you is that both are part of the equation, but neither explains it well enough to really be accurate.

From a slightly adapted previous post:

The single best explanation I've ever read about the creation of the Electoral College comes from Jack Rakove, who sums it up in a single sentence:

"The key to (its creation at) the (Constitutional) Convention was not that the Electoral College had great and unmistakable virtues, but that it had fewer perceived disadvantages than the leading alternatives."

As Alex Keyssar follows up,

"It was, in effect, a consensus second choice, made acceptable, in part, by the remarkably complex details of the electoral process, details that themselves constituted compromises among, or gestures toward, particular constituencies and convictions."

If you keep those two thoughts in mind, the explanation for how the mess of the Electoral College was created makes a lot more sense - in that yes, there was some conscious thought in giving smaller states and those benefiting from the three-fifths compromise some advantages, but that decision was more or less designed to be in line with all the other ones that had developed elsewhere during the Convention than anything particular to the selection of a President. In addition, the Founders were badly wrong in their guesses on how elections would work in practice, and their inability to predict this nearly blew up the nascent United States in 1800 and has plagued the electoral process ever since.

One thing that tends to get overlooked in discussions about turning over the mechanism of how to select a Chief Executive to the "Committee of Eleven" that came up with the Electoral College at the very end of the Convention in September 1787 is that it was not just put off until then - but that the delegates had tried and failed since May to come up with something workable, and that some of the most prominent Founders had admitted that it was the "most difficult" issue that they'd faced, and that it not only divided the Convention but as James Wilson put it, "will also divide the people out of doors."

The initial proposal was letting the legislature choose the Chief Executive for a single 7 year term, which was somewhat similar to how things had been done in Colonial times - having the Executive Branch be reliant on the Legislative for its powers. This didn't fare well with Gouvernour Morris and Elbridge Gerry ("usurpation and tyranny on the part of the Legislature"), and as we're still taught today in high school civics in the United States (at least I hope), James Madison went even farther - "Legislative, Executive, and Judiciary powers should be separately...and independently exercised."

At this point in July, the Convention spent more of its time working on Congressional representation (ending up with the 3/5ths Compromise between North and South and the Great Compromise between small and large states). When it kept getting brought up, there was starting to be consensus that the Executive selection should not be via Legislative selection, but how then could it be done? Elbridge Gerry felt that "the ignorance of the people (would make the election)...radically vicious", and there was no real talk about any theoretical natural right of the people to select their Executive - but that it shouldn't be left in the hands of the Legislature for the sake of efficacy and institutional balance. You then had all sorts of other objections to a direct election: that in an foreshadowing of the debates over the next 80 years on the role of government, that it would lead to a national government rather than a federal government, that it would be "impractical" (a catch-all term of opponents), that it would threaten the carefully crafted compromises on the Legislative side hashed out between free and slave states in July...take your pick. Despite Madison's strong support, numerous attempts proposing various types of direct election were voted down by as much as a 9-1 (it was by states) margin during this time, and some of the alternatives suggested were even worse: selection by governors of the states, election by 15 members of Congress selected by random lottery, allowing the people the right to vote for 3 candidates with restrictions by residency...the additional options were numerous, not particularly well thought out, and while the Convention periodically returned to the initial one of an executive selected by the legislature, it was unpopular enough so that delegates realized it was probably unworkable.

At the end of August the Convention finally outright gave up and turned the drafting of the selection process over to the Eleven (or as it was more formally known, "The Committee on Postponed Parts") - which notably included Madison, still favoring direct elections - and within a week they'd come up with something palatable, albeit only after one of the Eleven showed up late on the first day to find that the committee had yet again gone back to the original 7 year, legislative chosen proposal and tentatively agreed to it, but convinced them to open up debate again. It bypassed the arguments over large state/small state influence by simply defaulting to the proportional representation the Convention had already exhausted themselves on back in July, skipped over whether or not the people could be trusted by allowing states to determine how to choose electors themselves (selection by legislatures were the overwhelming choice until the 1830s, and as I've touched on elsewhere even as late as 1876, Colorado did so), and overall it was viewed as a clever, workable compromise by an exhausted convention.

In fact, the biggest debate over its adoption was not over the process it had come up with but on their recommendation on what would happen if an election hadn't produced a majority - since the initial proposal was that the Senate would choose the President in those cases. George Mason wasn't alone in feeling that a majority would rarely be reached (he predicted 19 out of 20 elections would not receive one) and the general consensus was that the Senate would be far too closely tied to the Executive under those circumstances - in which case the House choosing by state delegations was by far a better idea, both in the anticipated composition of that body's membership versus that of the Senate and in preserving all the balance of electoral compromises that had been reached, including small and slave states. Moreover, the delegates also added that electors couldn't be existing officeholders in the Federal government, presumably to isolate them a bit from the pressures of voting for someone they'd owe a patronage job to (while completely ineffective in implementation, actually a pretty farsighted attempt as patronage did in fact matter significantly throughout the 19th Century.)

So this is a very long way of saying the Electoral College was the best workable compromise they could come up with at the time. Unfortunately, many of its weaknesses were badly aggravated by the Founders being incredibly naive as to the formation of political parties influencing how the routine selection of anyone besides Washington would proceed. This process nearly blew up the nascent United States in the Election of 1800 - I talk a little about it here - and it's been an issue ever since.