As a non-USAian can someone explain if this has any basis in fact?
Not much.
Here's the relevant clause of the constitution
Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors.....etc.etc.
I think the vast majority of people, including Justices of the Supreme Court, would say what that means is the legislature dictates how elections will be run, and how electors will be appointed based on those elections, and, having done so, the state is required to stick with that law that was previously passed. In other words, the legislature can say that there will be a baseball game on election day, and the winner will get the electors. However, what they could not do would be to say that, once the game is going into extra innings, that we will stop the baseball game, and draw straws instead.
Having decide that they will play baseball, they have to stick to baseball.
In reality, all states say that there will be an election, and whoever gets the most votes gets the electors. (Maine and Nebraska have a slight wrinkle on that.) Where the states have some leeway is in electoral procedures. That's why in some states ballots have to be received at a certain time, whereas in other states, they have to be postmarked at a certain time.
What the crazy all caps Fox guy (Larry Elder?) was saying is that if there is some sort of question about how things are going, the legislature can step in and decide who gets the electors, and to heck with whatever law was passed before. That's a bit nutty.
On the other hand, suppose there was truly an irresolvable situation. Let's say it's a very close election, and a fire breaks out in a counting center, and a number of ballots that is sufficient to make the difference are destroyed before they are counted. What then?
I don't think the law has ever been tested on those extreme situations. I think that generally, it is left to state courts to interpret what the appropriate actions are based on the laws of the state, and occasionally by federal law as applicable. Sorting out specific situations is up to the lawyers and judges. However, it is conceivable that a legislature might pass some sort of law about how to sort it out, and it is conceivable a judge would accept the legislative solution, as long as it was perceived that the legislatures were trying to follow the spirit of the existing law, just in a situation where it's impossible to follow it directly.
So, the short answer is, no. He's wrong. The slightly longer answer is, no, he's wrong, but in an extreme situation he could be a little bit right.