If you believe that all votes matter, all voters are equal, and our country's leadership should be elected by the people, a national popular vote is the only way. Any system that allows the possibility of a "loser" to win defeats the goal of a democracy, which is to elect those who best represent the people's wishes.
A representative democracy is still a democracy.
Besides, this objection is meaningless because the argument being made is normative, not positive. Saying 'the U.S. is a republic, not a democracy' in response is an attempt to shut down discussion, not engage with it in a meaningful way.
As for this proposal, it is only marginally better than the existing system, and if there was enough will to make such an amendment it would be better to actually implement a modern voting system on the whole.
