TL;DR Amending that one piece won't fix the underlying issue and arguably exacerbates it.
The Senate used to have the same rules and at the start of the US some states had their state legislatures pick senators instead of holding a vote.
Also, electors from the vast majority of states are able to vote for whomever they want. Some of them will be fined if they don't vote for the candidate picked by the state, but only a few states outright ban electors from being faithless. The general election for President in November is only a non-binding poll. The real votes are counted on 1/6.
As an example, consider 2016:
| "Candidate" | Votes based on Nov. Poll | Actual Votes | Difference |
|---|---|---|---|
| Donald Trump | 306 | 304 | -2 |
| Hillary Clinton | 232 | 227 | -5 |
| Colin Powell | 0 | 3 | +3 |
| Faith Spotted Eagle | 0 | 1 | +1 |
| John Kasich | 0 | 1 | +1 |
| Ron Paul | 0 | 1 | +1 |
| Bernie Sanders | 0 | 1 | +1 |
The whole system, from the very existence of the Senate, to the 10th Amendment is designed to operate differently than it does today.
To drive the point home, I want to talk about the Civil War. I think it is a tragedy that everyone gets caught in a loop about it being fight over states' rights or slavery. Kids should be forced to read South Carolina's Declaration of the Immediate Causes which Induce and Justify the Secession of South Carolina from the Federal Union. The document explains very well how both sides of the argument are correct and, more importantly for our discussion, shines light on how the states viewed themselves when the federal government was established.
Here's a good excerpt
Under this Confederation the War of the Revolution was carried on, and on the 3d September, 1783, the contest ended, and a definite Treaty was signed by Great Britain, in which she acknowledged the Independence of the Colonies in the following terms:
“Article 1.— His Britannic Majesty acknowledges the said United States, viz: New Hampshire, Massachusetts Bay, Rhode Island and Providence Plantations, Connecticut, New York, New Jersey, Pennsylvania, Delaware, Maryland, Virginia, North Carolina, South Carolina and Georgia, to be FREE, SOVEREIGN AND INDEPENDENT STATES; that he treats with them as such; and for himself, his heirs and successors, relinquishes all claims to the government, propriety and territorial rights of the same and every part thereof.”
Thus were established the two great principles asserted by the Colonies, namely: the right of a State to govern itself; and the right of a people to abolish a Government when it becomes destructive of the ends for which it was instituted. And concurrent with the establishment of these principles, was the fact, that each Colony became and was recognized by the mother Country as a FREE, SOVEREIGN AND INDEPENDENT STATE.