I'm sorry but you are wrong. There are plenty of authoritative sources in addition to the BBC that explain this.
When the rocket lifts off, it is rotating at the same speed as the earth. It doesn't lose that immediately just because it is no longer in contact with the earth. If it goes straight up, it will remain directly over the launch pad.
And let us suppose for a moment that it did immediately lose its rotational speed. What would happen? The answer is that it would go UPrange, not DOWNrange. The earth spins west to east, so if the earth was rotating under the rocket it would go westwards, over mainland USA. It actually heads eastwards, out over the Atlantic Ocean. It is going the same way the earth is rotating, whereas if it was standing still and the earth was rotating under it, it would be going the opposite way to the earth's rotation. Heading east means it can add its velocity to the earth's rotational speed, helping it to get into orbit, and drop its boosters into the ocean, not onto land.
Artemis starts pitching fairly shortly after clearing the tower, so is around 8 miles downrange (i.e. over the Atlantic) after 1 minute, whereas if it was due to the earth's rotation it would be around 25km uprange. In contrast, Apollo didn't pitch over until later and was only 1 mile downrange after 1 minute. From the ground it looked pretty much like it was going straight up.
The BBC article is correct. The path looks curved because it is curved.