When they say “the people” they mean the people as individuals, not as a collective. For example, the 4th amendment says "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, "
“I, the People…”? Seriously?
We, as a group of individuals in a voluntary compact.
Piffle. We are monkeys. A bunch of monkeys is a tribe, which may or may not thrive. One monkey is a dead monkey.
That’s been understood since cavemen learned to bang rocks together. But how best to organize the tribe? Is top down rule any better when the leaders are elected? The founders didn’t think so. It’s not as if Britain was a tyranny where King George had absolute power, most of the edicts the colonists objected to came from Parliament. It wasn’t just the “taxation without representation” either. It was the whole doctrine that still exists today of Parliamentary Supremacy. Under Britain’s laws, Parliament can pass any law it wants on any subject it wants, without limitation. So the founders embraced a new idea: a limited, federalist state, with checks and balances, as opposed to the British model of a superlegislature that can do literally whatever it desires.
From the ever-reliable Wikipedia, but it’s sourced to Lawrence Tribe:
You’ll note I did not snip the last sentence. 
I don’t think it stretches the Compact Clause at all, and I think the proponents are suggesting congressional approval out of an abundance of caution. I guess if it passes, it’ll be up to the courts.
There’s also the question of whether the arrangement actually constitutes a “compact”. A declaration by X that X will do thus-and-such if and only if Y and Z act likewise does not necessarily require any bargain among X, Y, and Z.
“The people as individuals” is a contradiction in terms. It’s like saying that when they said “elephants”, they meant “small mammals”.
Then that would mean that the reference to “the people” in the 4th amendment makes protections from unreasonable search and seizure not an individual right.
Taken to the extreme, this would mean that two adjoining states that have bridges over their boundary rivers would have to go to Congress to approve an agreement to do any bridge rehabilitation. Or communities that straddle state lines would have to go to Congress to approve a common municipal water system. Or extradite each others’ prisoners. There are too many practical things that states do that Congress really doesn’t have a stake in, nor should it.
That being said, the popular vote compact is a disaster waiting to happen. Someday there will be an election where the national popular vote is tied to withn a thousand votes nationwide. A national recount would very well flip the result, as would a second recount and a third. Thus lies the value of the electoral college- it can give a definitive result where otherwise none might exist.
Highly unlikely to happen any time in the next century or so, by which point I would hope we’ve gotten a lot better at the whole counting votes thing.
Nor does it. The Supreme Court decided in Virginia v. Tennessee, 148 U.S. 503, that prior Congressional consent is required only when the agreement is:
Legal! And Constitutional! (We really got get that boy a “legal and Constitutional” smilie, save him a a lot of typing!)
Of course, the National Popular Vote Compact, if it ever kicked in, would actually tend to decrease the political power of the participating states.
Unlikely, but the kind of thing that happens when conditions are set up to make it as disastrous as possible. Murphy’s Law and all that.
It’s not a real problem if it’s planned for though. A national recount is out of the question. However, the compact could have states revert to the traditional system if the popular vote is within 1%.
Why can we not recount all the votes? It’s not really more complex than recounting the votes in a single precinct. It takes longer, sure. But you don’t have to count them one at a time.
I still think that the compact would require Congressional approval if for no other reason that the compact would render the EV’s of the other states meaningless. I’m sure that in this day and age that would be considered a Federal issue.
Because legally, you can’t demand that a state other than your own recount the votes. If California is part of a compact and they can’t figure out what to do in a close election, they can’t tell Texas, not in the compact, that they have to recount their votes when they’ve already satisfied their requirements.
This wouldn’t actually fix the problem: It’d just mean that instead of a recount to decide whether the vote was 49.997% or 50.003%, you’d need a recount to decide whether the vote was 48.997% or 49.003%. The actual solution would be that re-counts would go on until the Supreme Court ruled that they had to be stopped, and then we’d go with whatever the counts said at that time, just like the last time the fate of the Presidency was left uncertain due to a razor-thin margin. Which, please note, actually happened under the current system.
The compact would not render the EVs of the other states meaningless. The Electoral College renders them meaningless. The compact is an effort to remove that effect, not to institute it.