It would be very difficult to truly verify the national popular vote totals.
Consider the 2000 election, for example. On election night 2000, Bush was ahead in the national popular vote by about 300,000 votes. The next afternoon, as vote counting continued, the media announced that Al Gore was actually ahead in the popular vote, by about 500,000 votes. How did that happen.
(all of this ignores the Florida electoral vote recount. But the point is, under a national popular vote, the Florida recount wouldn’t have happened that way. There would have had to be a national recount instead.)
In 2000, did the Democrats pad the vote totals for Al Gore, to get him to be ahead in the mythical national popular vote, so they could claim some legitimacy as they pursued the Florida recount? It’s something to think about.
I live in Washington State where it is accepted practice for democrats to magically find boxes full of ballots for weeks after an election that they have not won. Eliminating the electoral college would guarantee their control on a national scale. Thompson should call his idea the ACORN amendment.