As much as I see your point of view, I would rather the majority of people's votes count much more than regional representation. In a weird way what you profess would actually mean one person's vote is more weighty than another's in densely populated areas. That isn't fair or democratic at all. I would rather have the person with the plurality of votes become President vs a plurality of states where a mind set is vastly different from the majority actually determines the outcome of an election. Conversely if you must keep the electoral college alive and well, then each state must only have the exact multiple of its population as its electoral vote tally. As an example, if the US population is 350MM and it counts for 538 Electoral college votes, then if a particular state has 35M people residing in it, then it should count for 54 Electoral college votes and so on and so forth. Just my thoughts on it if Electoral college must stay.
I'm actually a huge fan of the French system. In the first round anyone can run, and if no one gets 50% in that round (and iirc no one ever has) the top two go to the second round, where it's a straight majority. It means the candidates don't have to jostle for the extreme sides of their party to have a proper shot (like you saw with the Republicans in 2016 and you're seeing with the Democrats here). Then in the second round it's all about appealing to as many people as possible, so people who prove particularly divisive tend to get filtered out and the people whose candidates lost in the first round can't really complain or play "spoiler". As a result, candidates basically have to pass through two filters that deliberately work in different ways.
So in 2016 it might have been Bernie, Hillary, Donald, Jeb and Ted. Out of those five, you tend to get two or three stand out but they have to do it on their own merits against a more packed field. The top two out of those would then have to get as much of the country behind them as possible.
I understand the fear some have of a few cities dictating to the rest of the country but this is pretty overstated (not to mention that, like I've already said, the power of the president is pretty overstated as it is). The top 100 cities (including the mighty Spokane, WA - population 219,000) combined account for 19.8% of the population. The top 300 (now including Rialto, CA - which at population 105,000 is about the size of my home city, which gets laughed at even in New Zealand for being small) make up 28.6% of the vote. You can't win any kind of population-based vote by ignoring 81% of the electorate (nor can you win by ignoring 72% of it). But ironically enough, with the electoral college you can actually become president with only 22% of the vote if you focus on the right states. If you were to start your own country from scratch, there's no way you'd choose a system that allowed for a quirk like that.
If the goal is to get candidates to move around the country the results are (at best) questionable there too. I'll have to dig the research up again but when people have studied this, the data shows that the candidates basically spend all their time in Ohio and Florida plus a small handful of states where the vote may be close. And as you say (and
@52520Andrew says), if the goal is to not focus too much power in dense areas it's inadvertently or otherwise caused the opposite, where the vote of someone in Rhode Island counts for three votes of someone in Texas. Not to mention that the way the Senate is made up already acts as a check against proportional representation (and assuming that's a bad thing is already a pretty big leap).