No matter what method we have, someone will feel slighted. As you yourself pointed out, if we change the Constitution and eliminate the Electoral system, presidential candidates will concentrate their efforts on the four biggest states and ignore everyone else. The large states will be the new battleground states.
I don't see that your desired change will guarantee a larger voter turnout. All that will happen is voters won't turnout at different places. I.e. no one in small states like DE or WY will bother turning up to vote.
Of COURSE voters will turn out in small states. It's not about state size.
Support for a national popular vote has been strong in every smallest state surveyed in polls among Republicans, Democrats, and Independent voters, as well as every demographic group
Among the 13 lowest population states, the National Popular Vote bill has passed in 9 state legislative chambers, and been enacted by 4 jurisdictions.
More people register to vote and do vote when they know their vote matters.
If you're a Republican voter in a blue state or a Democratic voter in a red state, your vote for president doesn't matter to your candidate.
With the National Popular Vote bill in effect, presidential campaigns would poll, organize, visit, and appeal to more than 12 states. One would reasonably expect that voter turnout would rise in 70-80% of the country that is conceded months in advance by the minority parties in the states, taken for granted by the dominant party in the states, and ignored by all parties in presidential campaigns.
A national popular vote could increase down-ballot voter turnout during presidential election years.
Under National Popular Vote, every voter, everywhere, would be politically relevant and equal in every presidential election. Every vote would matter in the state counts and national count.
Voters in the minority in non-battleground states, red or blue, are cheated in every presidential election.
National Popular Vote would give a voice to the minority party voters in presidential elections in each state. Now they don't matter to their candidate.
In 2012, 56,256,178 (44%) of the 128,954,498 voters had their vote diverted by the winner-take-all rule to a candidate they opposed (namely, their state’s first-place candidate).
And now votes, beyond the one needed to get the most votes in the state, for winning in a state, are wasted and don't matter to candidates.
Oklahoma (7 electoral votes) alone generated a margin of 455,000 "wasted" votes for Bush in 2004 -- larger than the margin generated by the 9th and 10th largest states, namely New Jersey and North Carolina (each with 15 electoral votes).
Utah (5 electoral votes) alone generated a margin of 385,000 "wasted" votes for Bush in 2004.
8 small western states, with less than a third of California’s population, provided Bush with a bigger margin (1,283,076) than California provided Kerry (1,235,659).
In 2008, voter turnout in the then 15 battleground states averaged seven points higher than in the 35 non-battleground states.
In 2012, voter turnout was 11% higher in the then 9 battleground states than in the remainder of the country.
In 2016, in battleground states, turnout hit 65%, 5 points higher than in non-battleground states.