No they wouldn't. Politicians would only focus on major metropolitan areas - where the population density is highest. It's the biggest bang for their campaign buck. A purely popular vote for the president essentially means that LA and NYC determine how the country is led, with maybe a small bit of input from Chicago, Dallas, DC, and Philadelphia. Unless the entire rest of the US collaboratively fall son the other side, that's where the decision would get made. That might be nice for the democrats though - high density areas tend to be more liberal. Not sure that would be much mollification to Nebraska, Alaska, and the Dakotas though. Nothing like making absolutely certain that their voices don't matter, I guess...
Wrong. Even with the EC system, the Dems just paid that price... letting one candidate have free rein in districts, counties or states just increases their leads. The Dems gave up on middle class white Pennsylvania in hopes of running up the score in Philly. It was suicide. The hard line dem vote (black, largely) in Philly can only be as strong as the statewide demographic. There are only about 10% of Pennsylvanians who are black. Had Hillary's team not ignored the areas where she dropped 15/20% from Obama's totals - the rust belt areas in the west, she'd have won PA.
Similarly, in a straight popular vote you can double-down in New York City and Los Angeles, but count on the fact that Trump wouldn't have been ignoring them as much as he did. Clinton took NY State by 20%. Cutting that down by half by campaigning and spending money (the GOP spent nothing on Trump in NYS) would give Trump the popular vote.
Of course they'll campaign more heavily where there are more votes. But the guy or gal who gets out her cowboy boots and goes to the rodeo in August in Missoula, walks the range in Abilene or chomps on corn at the fair in Bismark is going to come out 5 to 10 thousand votes to the good in those states. And you do that ten times while your opponent is playing Madison Square Garden more than necessary, and you win the popular vote.
It would give greater attention to the flyover states, not less. You couldn't have a more-dedicated-to-ignore-you system than the first past the post EC system we currently have. Every election cycle X number of battlegrounds get the attention. This year poor old Virginia was left out in the cold. The last few cycles it was a battleground and had campaign visit after campaign visit. Last cycle in 2012, Pennsylvania got largely the same treatment except for a brief spurt when Rove convinced them they had a chance, there.
Presently everybody spends the bulk of their time in OH, FL, NC... add PA for this cycle and now add MI and the rust belt. Ain't no one campaigning in or addressing the local concerns in NY, CA, TX, IL. Big EC counts, but solidly belonging to one or the other. That's the way it works in a winner take all system. The thought that it'd be worse when every vote will actually count is rather absurd. They'd still spend all their money in the big markets, but they'd pay some attention to states they're bound to lose.
As I said... NY State. Trump and the GOP would not have allowed it to go +20 blue. Ditto CA. Montana, Wyoming, North Dakota, Idaho, etc... The Dems actually have some traditional support in those states. If Hillary or Bill or Biden had showed up there, she might cut their margins from +15 to +12. Do that often enough and you pick up another hundred thousand votes.