The EC can work but make it a contest for each electoral vote, and remove the states from the equation entirely. California being safe blue and Texas being safe red don't matter, each district is counted for one electoral vote, and the states don't get extra votes anymore.
That just seems like popular vote with extra steps. I'm not sure, but I feel like mathematically there would be no way in which the result of the EC would differ from the popular vote under such a system. I suppose it might still be possible to skew it far enough to shift the outcome using some extreme gerrymandering.
It is a popular vote with extra steps. That's literally what it is.
The extra steps mean that politicians can't purely focus on population centers, rural communities would count for the same vote. each district should be of similar population size, and every district counts for one.
Many democracies don't have the people directly vote on their leader. Parliamentary systems typically have the people voting for a representative who will then vote for the Prime Minister on their behalf.