Elections would be decided by strategies rather than votes. For simplicity's sake, let's assume that the campaigns could choose where their supporters voted. In reality, they could probably only encourage voter behavior, but let's think about it as if they had control.
Looking at 2016, the natural thought is that this would allow Hillary Clinton voters in a state like California to put their votes in more competitive states like Florida and North Carolina.
OK, but what happens when the Donald Trump campaign responds by asking some voters in California and New York to vote in New Hampshire, Minnesota, Nevada, Maine, Colorado, and Virginia? California may not have as many Republicans as Democrats, but Trump received more votes there than in all but two other states (Texas and Florida).
Another way of saying this is that Clinton's vote margin in California was smaller than Trump's total. I.e. she had less than double Trump's vote total in California. He could safely take all of his votes from there and put them in more competitive states. Meanwhile, she has to leave some of her votes there--she still wants to win the state.
Trump could also pull out of Utah, New Mexico, and Washington. Evan McMullin had enough votes nationwide to win Utah, and Gary Johnson could win New Mexico and Washington. Jill Stein might win Oregon. And that's just with the actual 2016 totals under the current system. Under this system, more might have voted for the third party candidates, as their votes could count.
It gets worse when we realize that voters won't necessarily vote as dictated. This makes it even riskier for Clinton to pull votes out of California. What if she pulls too many votes? She could potentially lose California. Meanwhile, Trump can safely pull votes out of California, New York, Hawaii, the District of Columbia, etc. States that he lost decisively.
Clinton's best bet would be to pull votes not out of California but out of Texas, West Virginia, North Dakota, and other states that she lost decisively.
Both campaigns would have to deal with the problem that they didn't actually win a majority of the popular vote. This is generally known about Trump, but was also true of Clinton. She only won 48.2% of the vote. While good strategy would give her a better chance of winning outright, neither campaign would be in good shape. They'd have a much higher chance of a tied result, which would have given the Republicans an advantage (they won the tiebreakers).
As a practical matter, the Trump campaign outperformed the Clinton campaign in allocating resources to states. They won three states that most people assumed that the Clinton campaign could rely on: Wisconsin, Michigan, and Pennsylvania. There is no reason to think that the Clintonites suddenly would have become better at strategy under this system. And bad strategy is more costly here.
There's also a turnout problem. In California, many Republicans had no candidate for whom to vote. Trump wasn't going to win the presidential vote in California. Republicans didn't even have a Senate candidate running, both candidates were Democrats. Many didn't have a legislative candidate running, because they were in overwhelmingly Democrat areas. Now those votes could count. Overall, that would favor Trump. He was blown out in more electoral college votes than Clinton was. Clinton was blown out in more states, but they had fewer electoral college votes.
Overall, this system would have favored the Republicans in 1996, 2000, 2004, 2012, and 2016 due to the House tiebreaker. In 2008, it would have favored the Democrats. That said, it would have increased the chances of a Democratic victory in 2000 and 2004. But it would have favored the Republicans in 1996 and 2012.
This system would increase the chances of the winner of a plurality or majority of the popular vote losing the election. This is because the strategies have a chance of working. Note that if you select perfectly while your opponents select badly, you can win with just eleven votes. If you put one vote each in the eleven biggest states, you will win 280 electoral college votes while your opponents will only win 258. That's an extremely unlikely case, but it shows what is possible.
Worse, this favors the popular vote loser. The loser can freely choose among strategies. Even a bad strategy doesn't actually cost the election. The popular vote winner will be expected to win, which will favor conservative strategies.
It would increase the chances of an electoral college tie going to the House for selection. And this favors third parties more. A third party candidate can relatively freely choose a strategy. They have no expectation of winning, so they can choose truly arbitrary strategies. A third party candidate could potentially win by making a deal with one of the major party candidates.
Currently, if someone wins a majority of the popular vote, they win the electoral college vote as well (with the singular exception of Samuel Tilden in 1876). It's only when no one wins the popular vote that the electoral college has a different result. In this system, there would be a small chance of the loser winning in every election. It's a better chance in a close election, but even in a landslide, it's possible for a good strategy to overcome a vote deficit.
FiveThirtyEight.com simulated a similar problem with effectively a hundred voters and ten states with one to ten electoral college votes each. No third parties and they gave both sides the same number of voters. The first place winner would lose to two of the top four strategies. In general, the first place winner only wins a little more than 80% of the time.
Here the system is much more complex. More voters. More states. More electoral college valuations. More parties.
Third party candidates could win states, making a tie more likely. The major party candidates would have to work out strategies. For example, Trump might move his votes from states like California and New York to states like Minnesota, New Hampshire, and Wisconsin. Good strategy could overcome a vote deficit. But the best strategy would depend on your opponent's strategy. For example, if Clinton pulled votes out of California, perhaps Trump should put votes in instead and go for the win.
Think Rock/Paper/Scissors where Rock sometimes beats paper. The popular vote winner would be less likely to win under this system.