The people shouldn't (directly) elect the President, because the office is a federal one; it should be the case that the states elect the President but it shouldn't be winner-takes-all in each state (except the two where it isn't, but still kinda is).
Adjusting the EC votes to be appropriated according to population, or even awarded based on the national popular vote outcome, is definitely an improvement. But the EC has vulnerabilities that were on display on Jan. 6th 2021.
The idea that the presidency serves the states is real, rooted in the founding of the US. But it is also somewhat antiquated in light of the civil war. Prior to the civil war, states definitely viewed themselves and the federal government in the way you're describing, with each state being its own little country with independent sovereignty. That idea was tested during the civil war, and the resolution was that the states are actually one nation, beholden to a federal government which unifies the nation. So leading up to the civil war, the states viewed themselves as, in some ways, higher in the pecking order than the federal government. The civil war made clear that it is the other way around.
In the years following the civil war, the idea of the federal government or the presidency serving the states has fallen away, and it has been replaced by the idea that the federal government serves the people directly in a unifying layer of government that the states play an organizational role within.
If you really step back and think about it, this was something of a required outcome, as too much state sovereignty can deeply erode the functioning of the nation. Not only did it threaten to break the nation due to a rights issue at the time (slavery), but it has already started to cut deep groves and erode commerce and travel, and create friction between state law enforcement in the US again over another rights issue (abortion). From this perspective, I think the notion that the EC is somehow essential to the functioning of the US or superior in some way to the popular vote has been upended by the civil war and is something that we're stuck with as a vestige. All it does at this point is create unrest, erode confidence in government, and create vulnerabilities for opportunists like Trump.
*It's worth noting that the EC and democracy itself played a role in the civil war. The south believed that they could not win elections, and it resulted in war. The same can be seen playing out today, as authoritarians grasp at ways to lock out democracy because of perceived shifts in the electability of certain concepts or parties. The "permanent minority" brand seems to immediately give way to fighting and disenfranchisement rather than to the soul-searching it should.
Edit: Somehow I wrote all of this without discussing the EC's role in slavery and the 3/5ths compromise. Not sure how I managed that. Notes for next time.